Theorem gsumzoppg 18344

Description: The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsumzoppg.b	⊢ 𝐵 = (Base‘𝐺)
gsumzoppg.0	⊢ 0 = (0g‘𝐺)
gsumzoppg.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzoppg.o	⊢ 𝑂 = (oppg‘𝐺)
gsumzoppg.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzoppg.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzoppg.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzoppg.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzoppg.n	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
gsumzoppg	⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

Proof of Theorem gsumzoppg

Dummy variables 𝑓 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzoppg.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		gsumzoppg.o	. . . . . . . . 9 ⊢ 𝑂 = (oppg‘𝐺)
3	2	oppgmnd 17784	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
4	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Mnd)
5		gsumzoppg.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		gsumzoppg.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
7	2, 6	oppgid 17786	. . . . . . . 8 ⊢ 0 = (0g‘𝑂)
8	7	gsumz 17374	. . . . . . 7 ⊢ ((𝑂 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	4, 5, 8	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	6	gsumz 17374	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
11	1, 5, 10	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
12	9, 11	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
13	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
14		gsumzoppg.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15		fvex 6201	. . . . . . . 8 ⊢ (0g‘𝐺) ∈ V
16	6, 15	eqeltri 2697	. . . . . . 7 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
18		ssid 3624	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))
19		fex 6490	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
20	14, 5, 19	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
21		suppimacnv 7306	. . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
22	20, 16, 21	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
23	22	sseq1d 3632	. . . . . . 7 ⊢ (𝜑 → ((𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })) ↔ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
24	18, 23	mpbiri 248	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
25	14, 5, 17, 24	gsumcllem 18309	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
26	25	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
27	25	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
28	13, 26, 27	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
29	28	ex 450	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
30		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
31		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
32	30, 31	syl6eleq 2711	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ (ℤ≥‘1))
33	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
34		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
35		dffn4 6121	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
36	34, 35	sylib 208	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
37		fof 6115	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–onto→ran 𝐹 → 𝐹:𝐴⟶ran 𝐹)
38	33, 36, 37	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
39	1	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
40		gsumzoppg.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
41	40	submacs 17365	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
42		acsmre 16313	. . . . . . . . . . . 12 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
43	39, 41, 42	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
44		eqid 2622	. . . . . . . . . . 11 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
45		frn 6053	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
46	33, 45	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ 𝐵)
47	43, 44, 46	mrcssidd 16285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
48	38, 47	fssd 6057	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
49		f1of1 6136	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })))
50	49	ad2antll 765	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })))
51		cnvimass 5485	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
52		fdm 6051	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
53	33, 52	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → dom 𝐹 = 𝐴)
54	51, 53	syl5sseq 3653	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
55		f1ss 6106	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧ (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
56	50, 54, 55	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
57		f1f 6101	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
58	56, 57	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
59		fco 6058	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
60	48, 58, 59	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
61	60	ffvelrnda 6359	. . . . . . 7 ⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(#‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐹 ∘ 𝑓)‘𝑥) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
62	44	mrccl 16271	. . . . . . . . . 10 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
63	43, 46, 62	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
64	2	oppgsubm 17792	. . . . . . . . 9 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
65	63, 64	syl6eleq 2711	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂))
66		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝑂) = (+g‘𝑂)
67	66	submcl 17353	. . . . . . . . 9 ⊢ ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
68	67	3expb 1266	. . . . . . . 8 ⊢ ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g‘𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
69	65, 68	sylan 488	. . . . . . 7 ⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g‘𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
70		gsumzoppg.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
71	70	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
72		gsumzoppg.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (Cntz‘𝐺)
73		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
74	72, 44, 73	cntzspan 18247	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
75	39, 71, 74	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
76	73, 72	submcmn2 18244	. . . . . . . . . . . . 13 ⊢ (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
77	63, 76	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
78	75, 77	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
79	78	sselda 3603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → 𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
80		eqid 2622	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
81	80, 72	cntzi 17762	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
82	79, 81	sylan 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
83	80, 2, 66	oppgplus 17779	. . . . . . . . 9 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
84	82, 83	syl6reqr 2675	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝑂)𝑦) = (𝑥(+g‘𝐺)𝑦))
85	84	anasss 679	. . . . . . 7 ⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g‘𝑂)𝑦) = (𝑥(+g‘𝐺)𝑦))
86	32, 61, 69, 85	seqfeq4 12850	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (seq1((+g‘𝑂), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))))
87	2, 40	oppgbas 17781	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
88		eqid 2622	. . . . . . 7 ⊢ (Cntz‘𝑂) = (Cntz‘𝑂)
89	39, 3	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
90	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
91	2, 72	oppgcntz 17794	. . . . . . . 8 ⊢ (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
92	71, 91	syl6sseq 3651	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((Cntz‘𝑂)‘ran 𝐹))
93		suppssdm 7308	. . . . . . . . . . . . . . 15 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
94	22, 93	syl6eqssr 3656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
95	94	adantl 482	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝜑) → (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
96		eqcom 2629	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐹 = 𝐴 ↔ 𝐴 = dom 𝐹)
97	96	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (dom 𝐹 = 𝐴 → 𝐴 = dom 𝐹)
98	97	adantr 481	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝜑) → 𝐴 = dom 𝐹)
99	95, 98	sseqtr4d 3642	. . . . . . . . . . . 12 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝜑) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
100	99	ex 450	. . . . . . . . . . 11 ⊢ (dom 𝐹 = 𝐴 → (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴))
101	52, 100	syl 17	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴))
102	14, 101	mpcom 38	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
103	102	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
104	50, 103, 55	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
105	23	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })) ↔ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
106	18, 105	mpbiri 248	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
107		f1ofo 6144	. . . . . . . . . . 11 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })))
108		forn 6118	. . . . . . . . . . 11 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
109	107, 108	syl 17	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
110	109	sseq2d 3633	. . . . . . . . 9 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
111	110	ad2antll 765	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
112	106, 111	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
113		eqid 2622	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
114	87, 7, 66, 88, 89, 90, 33, 92, 30, 104, 112, 113	gsumval3 18308	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (seq1((+g‘𝑂), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))))
115	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
116	115, 111	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
117	40, 6, 80, 72, 39, 90, 33, 71, 30, 104, 116, 113	gsumval3 18308	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐺 Σg 𝐹) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))))
118	86, 114, 117	3eqtr4d 2666	. . . . 5 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
119	118	expr 643	. . . 4 ⊢ ((𝜑 ∧ (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
120	119	exlimdv 1861	. . 3 ⊢ ((𝜑 ∧ (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
121	120	expimpd 629	. 2 ⊢ (𝜑 → (((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
122		gsumzoppg.n	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 )
123	122	fsuppimpd 8282	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
124	22, 123	eqeltrrd 2702	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
125		fz1f1o 14441	. . 3 ⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
126	124, 125	syl 17	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
127	29, 121, 126	mpjaod 396	1 ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   supp csupp 7295  Fincfn 7955   finSupp cfsupp 8275  1c1 9937  ℕcn 11020  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801  #chash 13117  Basecbs 15857   ↾_s cress 15858  +_gcplusg 15941  0_gc0g 16100   Σ_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245  Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748  opp_gcoppg 17775  CMndccmn 18193

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-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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-oppg 17776  df-cmn 18195

This theorem is referenced by:  gsumzinv  18345