Theorem islindf4 20177

Description: A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Hypotheses

Ref	Expression
islindf4.b	⊢ 𝐵 = (Base‘𝑊)
islindf4.r	⊢ 𝑅 = (Scalar‘𝑊)
islindf4.t	⊢ · = ( ·𝑠 ‘𝑊)
islindf4.z	⊢ 0 = (0g‘𝑊)
islindf4.y	⊢ 𝑌 = (0g‘𝑅)
islindf4.l	⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼))

Assertion

Ref	Expression
islindf4	⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌}))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐼   𝑥,𝐿   𝑥,𝑅   𝑥, ·   𝑥,𝑊   𝑥,𝑋   𝑥,𝑌   𝑥, 0

Proof of Theorem islindf4

Dummy variables 𝑗 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raldifsni 4324	. . . . 5 ⊢ (∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ (Base‘𝑅)((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌))
2		simpll1 1100	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ LMod)
3		simprll 802	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 ∈ (Base‘𝑅))
4		ffvelrn 6357	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝑗 ∈ 𝐼) → (𝐹‘𝑗) ∈ 𝐵)
5	4	3ad2antl3 1225	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (𝐹‘𝑗) ∈ 𝐵)
6	5	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐹‘𝑗) ∈ 𝐵)
7		islindf4.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝑊)
8		islindf4.r	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅 = (Scalar‘𝑊)
9		islindf4.t	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
10		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (invg‘𝑊) = (invg‘𝑊)
11		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (invg‘𝑅) = (invg‘𝑅)
12		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝑅)
13	7, 8, 9, 10, 11, 12	lmodvsinv 19036	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹‘𝑗) ∈ 𝐵) → (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = ((invg‘𝑊)‘(𝑙 · (𝐹‘𝑗))))
14	2, 3, 6, 13	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = ((invg‘𝑊)‘(𝑙 · (𝐹‘𝑗))))
15	14	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((invg‘𝑊)‘(𝑙 · (𝐹‘𝑗))) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))))
16		lmodgrp 18870	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
17	2, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Grp)
18	7, 8, 9, 12	lmodvscl 18880	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹‘𝑗) ∈ 𝐵) → (𝑙 · (𝐹‘𝑗)) ∈ 𝐵)
19	2, 3, 6, 18	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑙 · (𝐹‘𝑗)) ∈ 𝐵)
20		islindf4.z	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0g‘𝑊)
21		lmodcmn 18911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
22	2, 21	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ CMnd)
23		simpll2 1101	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼 ∈ 𝑋)
24		difexg 4808	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ 𝑋 → (𝐼 ∖ {𝑗}) ∈ V)
25	23, 24	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐼 ∖ {𝑗}) ∈ V)
26		simprlr 803	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))
27		elmapi 7879	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → 𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅))
28	26, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅))
29		simpll3 1102	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹:𝐼⟶𝐵)
30		difss 3737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∖ {𝑗}) ⊆ 𝐼
31		fssres 6070	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐼⟶𝐵 ∧ (𝐼 ∖ {𝑗}) ⊆ 𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
32	29, 30, 31	sylancl 694	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
33	8, 12, 9, 7, 2, 28, 32, 25	lcomf 18902	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))):(𝐼 ∖ {𝑗})⟶𝐵)
34		islindf4.y	. . . . . . . . . . . . . . . . 17 ⊢ 𝑌 = (0g‘𝑅)
35		simprr 796	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 finSupp 𝑌)
36	8, 12, 9, 7, 2, 28, 32, 25, 20, 34, 35	lcomfsupp 18903	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))) finSupp 0 )
37	7, 20, 22, 25, 33, 36	gsumcl 18316	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ∈ 𝐵)
38		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑊) = (+g‘𝑊)
39	7, 38, 20, 10	grpinvid2 17471	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Grp ∧ (𝑙 · (𝐹‘𝑗)) ∈ 𝐵 ∧ (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ∈ 𝐵) → (((invg‘𝑊)‘(𝑙 · (𝐹‘𝑗))) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g‘𝑊)(𝑙 · (𝐹‘𝑗))) = 0 ))
40	17, 19, 37, 39	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((invg‘𝑊)‘(𝑙 · (𝐹‘𝑗))) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g‘𝑊)(𝑙 · (𝐹‘𝑗))) = 0 ))
41		simplr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗 ∈ 𝐼)
42		fsnunf2 6452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) ∧ 𝑗 ∈ 𝐼 ∧ 𝑙 ∈ (Base‘𝑅)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
43	28, 41, 3, 42	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
44	8, 12, 9, 7, 2, 43, 29, 23	lcomf 18902	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹):𝐼⟶𝐵)
45		simpr 477	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
46		simpl 473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) → 𝑙 ∈ (Base‘𝑅))
47	45, 46	anim12i 590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑗 ∈ 𝐼 ∧ 𝑙 ∈ (Base‘𝑅)))
48		elmapfun 7881	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → Fun 𝑦)
49		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → dom 𝑦 = (𝐼 ∖ {𝑗}))
50		neldifsnd 4322	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (dom 𝑦 = (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ (𝐼 ∖ {𝑗}))
51		df-nel 2898	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ dom 𝑦)
52		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦 ↔ 𝑗 ∈ (𝐼 ∖ {𝑗})))
53	52	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑦 = (𝐼 ∖ {𝑗}) → (¬ 𝑗 ∈ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
54	51, 53	syl5bb 272	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
55	50, 54	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (dom 𝑦 = (𝐼 ∖ {𝑗}) → 𝑗 ∉ dom 𝑦)
56	49, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → 𝑗 ∉ dom 𝑦)
57	27, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → 𝑗 ∉ dom 𝑦)
58	48, 57	jca 554	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → (Fun 𝑦 ∧ 𝑗 ∉ dom 𝑦))
59	58	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) → (Fun 𝑦 ∧ 𝑗 ∉ dom 𝑦))
60	59	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (Fun 𝑦 ∧ 𝑗 ∉ dom 𝑦))
61	47, 60	jca 554	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑗 ∈ 𝐼 ∧ 𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦 ∧ 𝑗 ∉ dom 𝑦)))
62		funsnfsupp 8299	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ 𝐼 ∧ 𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦 ∧ 𝑗 ∉ dom 𝑦)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 ↔ 𝑦 finSupp 𝑌))
63	62	bicomd 213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ 𝐼 ∧ 𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦 ∧ 𝑗 ∉ dom 𝑦)) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
64	61, 63	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
65	64	biimpd 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
66	65	impr 649	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌)
67	8, 12, 9, 7, 2, 43, 29, 23, 20, 34, 66	lcomfsupp 18903	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) finSupp 0 )
68		incom 3805	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ({𝑗} ∩ (𝐼 ∖ {𝑗}))
69		disjdif 4040	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑗} ∩ (𝐼 ∖ {𝑗})) = ∅
70	68, 69	eqtri 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅
71	70	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅)
72		difsnid 4341	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ 𝐼 → ((𝐼 ∖ {𝑗}) ∪ {𝑗}) = 𝐼)
73	72	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ 𝐼 → 𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
74	41, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
75	7, 20, 38, 22, 23, 44, 67, 71, 74	gsumsplit 18328	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = ((𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})))(+g‘𝑊)(𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}))))
76		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
77		snex 4908	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {⟨𝑗, 𝑙⟩} ∈ V
78	76, 77	unex 6956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V
79		simpl3 1066	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → 𝐹:𝐼⟶𝐵)
80		simpl2 1065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → 𝐼 ∈ 𝑋)
81		fex 6490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ 𝑋) → 𝐹 ∈ V)
82	79, 80, 81	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → 𝐹 ∈ V)
83	82	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 ∈ V)
84		offres 7163	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V ∧ 𝐹 ∈ V) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
85	78, 83, 84	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
86		ffn 6045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → 𝑦 Fn (𝐼 ∖ {𝑗}))
87	28, 86	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 Fn (𝐼 ∖ {𝑗}))
88		neldifsn 4321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})
89		fsnunres 6454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 Fn (𝐼 ∖ {𝑗}) ∧ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
90	87, 88, 89	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
91	90	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))) = (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
92	85, 91	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
93	92	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗}))) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))))
94		ffn 6045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹):𝐼⟶𝐵 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼)
95	44, 94	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼)
96		fnressn 6425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼 ∧ 𝑗 ∈ 𝐼) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩})
97	95, 41, 96	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩})
98		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
99	43, 98	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
100		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
101	29, 100	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 Fn 𝐼)
102		fnfvof 6911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼 ∧ 𝐹 Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑗 ∈ 𝐼)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹‘𝑗)))
103	99, 101, 23, 41, 102	syl22anc 1327	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹‘𝑗)))
104		fndm 5990	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 Fn (𝐼 ∖ {𝑗}) → dom 𝑦 = (𝐼 ∖ {𝑗}))
105	104	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 Fn (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦 ↔ 𝑗 ∈ (𝐼 ∖ {𝑗})))
106	88, 105	mtbiri 317	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 Fn (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ dom 𝑦)
107		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑗 ∈ V
108		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑙 ∈ V
109		fsnunfv 6453	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑗 ∈ V ∧ 𝑙 ∈ V ∧ ¬ 𝑗 ∈ dom 𝑦) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
110	107, 108, 109	mp3an12 1414	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑗 ∈ dom 𝑦 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
111	87, 106, 110	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
112	111	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹‘𝑗)) = (𝑙 · (𝐹‘𝑗)))
113	103, 112	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (𝑙 · (𝐹‘𝑗)))
114	113	opeq2d 4409	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩ = ⟨𝑗, (𝑙 · (𝐹‘𝑗))⟩)
115	114	sneqd 4189	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩} = {⟨𝑗, (𝑙 · (𝐹‘𝑗))⟩})
116		ovex 6678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 · (𝐹‘𝑗)) ∈ V
117		fmptsn 6433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ V ∧ (𝑙 · (𝐹‘𝑗)) ∈ V) → {⟨𝑗, (𝑙 · (𝐹‘𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗))))
118	107, 116, 117	mp2an 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {⟨𝑗, (𝑙 · (𝐹‘𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗)))
119	118	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → {⟨𝑗, (𝑙 · (𝐹‘𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗))))
120	97, 115, 119	3eqtrd 2660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗))))
121	120	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗})) = (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗)))))
122		cmnmnd 18208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ CMnd → 𝑊 ∈ Mnd)
123	22, 122	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Mnd)
124	107	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗 ∈ V)
125		eqidd 2623	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑗 → (𝑙 · (𝐹‘𝑗)) = (𝑙 · (𝐹‘𝑗)))
126	7, 125	gsumsn 18354	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Mnd ∧ 𝑗 ∈ V ∧ (𝑙 · (𝐹‘𝑗)) ∈ 𝐵) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗)))) = (𝑙 · (𝐹‘𝑗)))
127	123, 124, 19, 126	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹‘𝑗)))) = (𝑙 · (𝐹‘𝑗)))
128	121, 127	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗})) = (𝑙 · (𝐹‘𝑗)))
129	93, 128	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})))(+g‘𝑊)(𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}))) = ((𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g‘𝑊)(𝑙 · (𝐹‘𝑗))))
130	75, 129	eqtr2d 2657	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g‘𝑊)(𝑙 · (𝐹‘𝑗))) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)))
131	130	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g‘𝑊)(𝑙 · (𝐹‘𝑗))) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
132	15, 40, 131	3bitrd 294	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
133	111	eqcomd 2628	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
134	133	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑙 = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
135	132, 134	imbi12d 334	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
136	135	anassrs 680	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) ∧ 𝑦 finSupp 𝑌) → (((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
137	136	pm5.74da 723	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑦 finSupp 𝑌 → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌)) ↔ (𝑦 finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
138		impexp 462	. . . . . . . . . . 11 ⊢ (((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (𝑦 finSupp 𝑌 → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌)))
139	138	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (𝑦 finSupp 𝑌 → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌))))
140	64	bicomd 213	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 ↔ 𝑦 finSupp 𝑌))
141	140	imbi1d 331	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)) ↔ (𝑦 finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
142	137, 139, 141	3bitr4d 300	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
143	142	2ralbidva 2988	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
144		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
145		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥 ∘𝑓 · 𝐹) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹))
146	145	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)))
147	146	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
148		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥‘𝑗) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
149	148	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥‘𝑗) = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
150	147, 149	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
151	144, 150	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)) ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
152	151	ralxpmap 7907	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝐼 → (∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
153	152	adantl 482	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
154	143, 153	bitr4d 271	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌))))
155		breq1 4656	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 finSupp 𝑌 ↔ 𝑥 finSupp 𝑌))
156	155	ralrab 3368	. . . . . . 7 ⊢ (∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌) ↔ ∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
157	154, 156	syl6bbr 278	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
158		resima 5431	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})) = (𝐹 “ (𝐼 ∖ {𝑗}))
159	158	eqcomi 2631	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝐼 ∖ {𝑗})) = ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))
160	159	fveq2i 6194	. . . . . . . . . . 11 ⊢ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) = ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})))
161	160	eleq2i 2693	. . . . . . . . . 10 ⊢ ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))))
162		eqid 2622	. . . . . . . . . . 11 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
163	79, 30, 31	sylancl 694	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
164		simpl1 1064	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → 𝑊 ∈ LMod)
165	24	3ad2ant2 1083	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐼 ∖ {𝑗}) ∈ V)
166	165	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (𝐼 ∖ {𝑗}) ∈ V)
167	162, 7, 12, 8, 34, 9, 163, 164, 166	ellspd 20141	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))) ↔ ∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))))))
168	161, 167	syl5bb 272	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → ((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))))))
169	168	imbi1d 331	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ (∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
170		r19.23v 3023	. . . . . . . 8 ⊢ (∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌))
171	169, 170	syl6bbr 278	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
172	171	ralbidv 2986	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑙 ∈ (Base‘𝑅)((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) = (𝑊 Σg (𝑦 ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
173		fvex 6201	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) ∈ V
174	8, 173	eqeltri 2697	. . . . . . . . . . 11 ⊢ 𝑅 ∈ V
175		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
176		eqid 2622	. . . . . . . . . . . 12 ⊢ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌}
177	175, 12, 34, 176	frlmbas 20099	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ 𝑋) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
178	174, 177	mpan 706	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑋 → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
179	178	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
180	179	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
181		islindf4.l	. . . . . . . 8 ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼))
182	180, 181	syl6reqr 2675	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → 𝐿 = {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌})
183	182	raleqdv 3144	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌) ↔ ∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
184	157, 172, 183	3bitr4d 300	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑙 ∈ (Base‘𝑅)((((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
185	1, 184	syl5bb 272	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
186	8	lmodfgrp 18872	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Grp)
187	12, 34, 11	grpinvnzcl 17487	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg‘𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
188	186, 187	sylan 488	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg‘𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
189	12, 34, 11	grpinvnzcl 17487	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg‘𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
190	186, 189	sylan 488	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg‘𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
191		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) → 𝑘 ∈ (Base‘𝑅))
192	12, 11	grpinvinv 17482	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝑘 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘((invg‘𝑅)‘𝑘)) = 𝑘)
193	186, 191, 192	syl2an 494	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg‘𝑅)‘((invg‘𝑅)‘𝑘)) = 𝑘)
194	193	eqcomd 2628	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → 𝑘 = ((invg‘𝑅)‘((invg‘𝑅)‘𝑘)))
195		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑙 = ((invg‘𝑅)‘𝑘) → ((invg‘𝑅)‘𝑙) = ((invg‘𝑅)‘((invg‘𝑅)‘𝑘)))
196	195	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑙 = ((invg‘𝑅)‘𝑘) → (𝑘 = ((invg‘𝑅)‘𝑙) ↔ 𝑘 = ((invg‘𝑅)‘((invg‘𝑅)‘𝑘))))
197	196	rspcev 3309	. . . . . . . 8 ⊢ ((((invg‘𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}) ∧ 𝑘 = ((invg‘𝑅)‘((invg‘𝑅)‘𝑘))) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg‘𝑅)‘𝑙))
198	190, 194, 197	syl2anc 693	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg‘𝑅)‘𝑙))
199		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑘 = ((invg‘𝑅)‘𝑙) → (𝑘 · (𝐹‘𝑗)) = (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)))
200	199	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑘 = ((invg‘𝑅)‘𝑙) → ((𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
201	200	notbid 308	. . . . . . . 8 ⊢ (𝑘 = ((invg‘𝑅)‘𝑙) → (¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
202	201	adantl 482	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 = ((invg‘𝑅)‘𝑙)) → (¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
203	188, 198, 202	ralxfrd 4879	. . . . . 6 ⊢ (𝑊 ∈ LMod → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
204	203	3ad2ant1 1082	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
205	204	adantr 481	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg‘𝑅)‘𝑙) · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
206		simplr 792	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝐿) → 𝑗 ∈ 𝐼)
207		fvex 6201	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
208	34, 207	eqeltri 2697	. . . . . . . . 9 ⊢ 𝑌 ∈ V
209	208	fvconst2 6469	. . . . . . . 8 ⊢ (𝑗 ∈ 𝐼 → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
210	206, 209	syl 17	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝐿) → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
211	210	eqeq2d 2632	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝐿) → ((𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗) ↔ (𝑥‘𝑗) = 𝑌))
212	211	imbi2d 330	. . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝐿) → (((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
213	212	ralbidva 2985	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = 𝑌)))
214	185, 205, 213	3bitr4d 300	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑗 ∈ 𝐼) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗))))
215	214	ralbidva 2985	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑗 ∈ 𝐼 ∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗))))
216	7, 9, 162, 8, 12, 34	islindf2 20153	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑗 ∈ 𝐼 ∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹‘𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
217	175, 12, 181	frlmbasf 20104	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
218	217	3ad2antl2 1224	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑥 ∈ 𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
219		ffn 6045	. . . . . . 7 ⊢ (𝑥:𝐼⟶(Base‘𝑅) → 𝑥 Fn 𝐼)
220	218, 219	syl 17	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑥 ∈ 𝐿) → 𝑥 Fn 𝐼)
221		fnconstg 6093	. . . . . . 7 ⊢ (𝑌 ∈ V → (𝐼 × {𝑌}) Fn 𝐼)
222	208, 221	ax-mp 5	. . . . . 6 ⊢ (𝐼 × {𝑌}) Fn 𝐼
223		eqfnfv 6311	. . . . . 6 ⊢ ((𝑥 Fn 𝐼 ∧ (𝐼 × {𝑌}) Fn 𝐼) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)))
224	220, 222, 223	sylancl 694	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)))
225	224	imbi2d 330	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝑥 ∈ 𝐿) → (((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌})) ↔ ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗))))
226	225	ralbidva 2985	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗))))
227		r19.21v 2960	. . . . 5 ⊢ (∀𝑗 ∈ 𝐼 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)))
228	227	ralbii 2980	. . . 4 ⊢ (∀𝑥 ∈ 𝐿 ∀𝑗 ∈ 𝐼 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)))
229		ralcom 3098	. . . 4 ⊢ (∀𝑥 ∈ 𝐿 ∀𝑗 ∈ 𝐼 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)))
230	228, 229	bitr3i 266	. . 3 ⊢ (∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗)))
231	226, 230	syl6bb 276	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → (𝑥‘𝑗) = ((𝐼 × {𝑌})‘𝑗))))
232	215, 216, 231	3bitr4d 300	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  dom cdm 5114   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ↑_𝑚 cmap 7857   finSupp cfsupp 8275  Basecbs 15857  +_gcplusg 15941  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100   Σ_g cgsu 16101  Mndcmnd 17294  Grpcgrp 17422  inv_gcminusg 17423  CMndccmn 18193  LModclmod 18863  LSpanclspn 18971   freeLMod cfrlm 20090   LIndF clindf 20143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022  df-lbs 19075  df-sra 19172  df-rgmod 19173  df-nzr 19258  df-dsmm 20076  df-frlm 20091  df-uvc 20122  df-lindf 20145

This theorem is referenced by:  islindf5  20178  matunitlindflem1  33405  aacllem  42547