Theorem psrass1 19405

Description: Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)

Hypotheses

Ref	Expression
psrring.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrring.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrring.r	⊢ (𝜑 → 𝑅 ∈ Ring)
psrass.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
psrass.t	⊢ × = (.r‘𝑆)
psrass.b	⊢ 𝐵 = (Base‘𝑆)
psrass.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
psrass.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
psrass.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
psrass1	⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Distinct variable groups:   𝑓,𝐼   𝑅,𝑓   𝑓,𝑋   𝑓,𝑍   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐷(𝑓)   𝑆(𝑓)   × (𝑓)   𝑉(𝑓)

Proof of Theorem psrass1

Dummy variables 𝑥 𝑘 𝑧 𝑔 ℎ 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrring.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2622	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrass.d	. . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
4		psrass.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
5		psrass.t	. . . . 5 ⊢ × = (.r‘𝑆)
6		psrring.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		psrass.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		psrass.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 4, 5, 6, 7, 8	psrmulcl 19388	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10		psrass.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	1, 4, 5, 6, 9, 10	psrmulcl 19388	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
12	1, 2, 3, 4, 11	psrelbas 19379	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
13	12	ffnd 6046	. 2 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
14	1, 4, 5, 6, 8, 10	psrmulcl 19388	. . . . 5 ⊢ (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
15	1, 4, 5, 6, 7, 14	psrmulcl 19388	. . . 4 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
16	1, 2, 3, 4, 15	psrelbas 19379	. . 3 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
17	16	ffnd 6046	. 2 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18		eqid 2622	. . . . 5 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}
19		psrring.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐼 ∈ 𝑉)
21		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
22		ringcmn 18581	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
23	6, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
24	23	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd)
25	6	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑅 ∈ Ring)
26	25	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑅 ∈ Ring)
27	1, 2, 3, 4, 7	psrelbas 19379	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
28	27	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
29		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
30		breq1 4656	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑗 ∘𝑟 ≤ 𝑥))
31	30	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑥))
32	29, 31	sylib 208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑥))
33	32	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ 𝐷)
34	28, 33	ffvelrnd 6360	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
35	34	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
36	1, 2, 3, 4, 8	psrelbas 19379	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
37	36	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
38		simpr 477	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)})
39		breq1 4656	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗) ↔ 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
40	39	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↔ (𝑛 ∈ 𝐷 ∧ 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
41	38, 40	sylib 208	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → (𝑛 ∈ 𝐷 ∧ 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
42	41	simpld 475	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑛 ∈ 𝐷)
43	37, 42	ffvelrnd 6360	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → (𝑌‘𝑛) ∈ (Base‘𝑅))
44	1, 2, 3, 4, 10	psrelbas 19379	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:𝐷⟶(Base‘𝑅))
45	44	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
46	19	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝐼 ∈ 𝑉)
47	46	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝐼 ∈ 𝑉)
48		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
49	3	psrbagf 19365	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐷) → 𝑗:𝐼⟶ℕ0)
50	46, 33, 49	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
51	32	simprd 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∘𝑟 ≤ 𝑥)
52	3	psrbagcon 19371	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘𝑟 ≤ 𝑥)) → ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑥))
53	46, 48, 50, 51, 52	syl13anc 1328	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑥))
54	53	simpld 475	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷)
55	54	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷)
56	3	psrbagf 19365	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑛 ∈ 𝐷) → 𝑛:𝐼⟶ℕ0)
57	47, 42, 56	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑛:𝐼⟶ℕ0)
58	41	simprd 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗))
59	3	psrbagcon 19371	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ 𝑛:𝐼⟶ℕ0 ∧ 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗))) → (((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
60	47, 55, 57, 58, 59	syl13anc 1328	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → (((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
61	60	simpld 475	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) ∈ 𝐷)
62	45, 61	ffvelrnd 6360	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)) ∈ (Base‘𝑅))
63		eqid 2622	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
64	2, 63	ringcl 18561	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑌‘𝑛) ∈ (Base‘𝑅) ∧ (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)) ∈ (Base‘𝑅)) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))) ∈ (Base‘𝑅))
65	26, 43, 62, 64	syl3anc 1326	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))) ∈ (Base‘𝑅))
66	2, 63	ringcl 18561	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑗) ∈ (Base‘𝑅) ∧ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))) ∈ (Base‘𝑅)) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ (Base‘𝑅))
67	26, 35, 65, 66	syl3anc 1326	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ (Base‘𝑅))
68	67	anasss 679	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)})) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ (Base‘𝑅))
69		fveq2 6191	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → (𝑌‘𝑛) = (𝑌‘(𝑘 ∘𝑓 − 𝑗)))
70		oveq2 6658	. . . . . . . 8 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) = ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)))
71	70	fveq2d 6195	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)) = (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))
72	69, 71	oveq12d 6668	. . . . . 6 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))) = ((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)))))
73	72	oveq2d 6666	. . . . 5 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))))
74	3, 18, 20, 21, 2, 24, 68, 73	psrass1lem 19377	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))))))))
75	7	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑋 ∈ 𝐵)
76	8	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑌 ∈ 𝐵)
77		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
78		breq1 4656	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑘 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑘 ∘𝑟 ≤ 𝑥))
79	78	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘𝑟 ≤ 𝑥))
80	77, 79	sylib 208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘𝑟 ≤ 𝑥))
81	80	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ 𝐷)
82	1, 4, 63, 5, 3, 75, 76, 81	psrmulval 19386	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗))))))
83	82	oveq1d 6665	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))
84		eqid 2622	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
85		eqid 2622	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
86	6	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑅 ∈ Ring)
87	19	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝐼 ∈ 𝑉)
88	3	psrbaglefi 19372	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ∈ Fin)
89	87, 81, 88	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ∈ Fin)
90	44	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
91		simplr 792	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
92	3	psrbagf 19365	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0)
93	87, 81, 92	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
94	80	simprd 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∘𝑟 ≤ 𝑥)
95	3	psrbagcon 19371	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑘:𝐼⟶ℕ0 ∧ 𝑘 ∘𝑟 ≤ 𝑥)) → ((𝑥 ∘𝑓 − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑘) ∘𝑟 ≤ 𝑥))
96	87, 91, 93, 94, 95	syl13anc 1328	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑘) ∘𝑟 ≤ 𝑥))
97	96	simpld 475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑘) ∈ 𝐷)
98	90, 97	ffvelrnd 6360	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑍‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅))
99	86	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring)
100	27	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
101		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘})
102		breq1 4656	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑗 → (ℎ ∘𝑟 ≤ 𝑘 ↔ 𝑗 ∘𝑟 ≤ 𝑘))
103	102	elrab 3363	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑘))
104	101, 103	sylib 208	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑘))
105	104	simpld 475	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ 𝐷)
106	100, 105	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
107	36	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
108	87	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑉)
109	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷)
110	108, 105, 49	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗:𝐼⟶ℕ0)
111	104	simprd 479	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 ∘𝑟 ≤ 𝑘)
112	3	psrbagcon 19371	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑘 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘𝑟 ≤ 𝑘)) → ((𝑘 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑘))
113	108, 109, 110, 111, 112	syl13anc 1328	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑘))
114	113	simpld 475	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ 𝐷)
115	107, 114	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑌‘(𝑘 ∘𝑓 − 𝑗)) ∈ (Base‘𝑅))
116	2, 63	ringcl 18561	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘𝑓 − 𝑗)) ∈ (Base‘𝑅)) → ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗))) ∈ (Base‘𝑅))
117	99, 106, 115, 116	syl3anc 1326	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗))) ∈ (Base‘𝑅))
118		eqid 2622	. . . . . . . . 9 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗))))
119		fvex 6201	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
120	119	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (0g‘𝑅) ∈ V)
121	118, 89, 117, 120	fsuppmptdm 8286	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))) finSupp (0g‘𝑅))
122	2, 84, 85, 63, 86, 89, 98, 117, 121	gsummulc1 18606	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))
123	98	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑍‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅))
124	2, 63	ringass 18564	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((𝑋‘𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘𝑓 − 𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅))) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘)))))
125	99, 106, 115, 123, 124	syl13anc 1328	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘)))))
126	3	psrbagf 19365	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
127	19, 126	sylan 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
128	127	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0)
129	128	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
130	93	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
131	130	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
132	110	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
133		nn0cn 11302	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
134		nn0cn 11302	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
135		nn0cn 11302	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
136		nnncan2 10318	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑘‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
137	133, 134, 135, 136	syl3an 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
138	129, 131, 132, 137	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
139	138	mpteq2dva 4744	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧))))
140		ovexd 6680	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V)
141		ovexd 6680	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V)
142	128	feqmptd 6249	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
143	110	feqmptd 6249	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
144	108, 129, 132, 142, 143	offval2 6914	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑥 ∘𝑓 − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑗‘𝑧))))
145	130	feqmptd 6249	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
146	108, 131, 132, 145, 143	offval2 6914	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑗‘𝑧))))
147	108, 140, 141, 144, 146	offval2 6914	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)) = (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))))
148	108, 129, 131, 142, 145	offval2 6914	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑥 ∘𝑓 − 𝑘) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧))))
149	139, 147, 148	3eqtr4d 2666	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)) = (𝑥 ∘𝑓 − 𝑘))
150	149	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))) = (𝑍‘(𝑥 ∘𝑓 − 𝑘)))
151	150	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → ((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)))) = ((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))
152	151	oveq2d 6666	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘)))))
153	125, 152	eqtr4d 2659	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))))
154	153	mpteq2dva 4744	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)))))))
155	154	oveq2d 6666	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))))))
156	83, 122, 155	3eqtr2d 2662	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))))))
157	156	mpteq2dva 4744	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘)))) = (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)))))))))
158	157	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))))))))
159	8	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑌 ∈ 𝐵)
160	10	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑍 ∈ 𝐵)
161	1, 4, 63, 5, 3, 159, 160, 54	psrmulval 19386	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗)) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))))))
162	161	oveq2d 6666	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))))))
163	3	psrbaglefi 19372	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ Fin)
164	46, 54, 163	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ Fin)
165		ovex 6678	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
166	3, 165	rab2ex 4816	. . . . . . . . . . . 12 ⊢ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ V
167	166	mptex 6486	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ V
168		funmpt 5926	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))))
169	167, 168, 119	3pm3.2i 1239	. . . . . . . . . 10 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∧ (0g‘𝑅) ∈ V)
170	169	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∧ (0g‘𝑅) ∈ V))
171		suppssdm 7308	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) supp (0g‘𝑅)) ⊆ dom (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))))
172		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) = (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))))
173	172	dmmptss 5631	. . . . . . . . . . 11 ⊢ dom (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}
174	171, 173	sstri 3612	. . . . . . . . . 10 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}
175	174	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)})
176		suppssfifsupp 8290	. . . . . . . . 9 ⊢ ((((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∧ (0g‘𝑅) ∈ V) ∧ ({ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ Fin ∧ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)})) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) finSupp (0g‘𝑅))
177	170, 164, 175, 176	syl12anc 1324	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) finSupp (0g‘𝑅))
178	2, 84, 85, 63, 25, 164, 34, 65, 177	gsummulc2 18607	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))))))
179	162, 178	eqtr4d 2659	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗))) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))))))
180	179	mpteq2dva 4744	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗)))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))))))))
181	180	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))))))))
182	74, 158, 181	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗))))))
183	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
184	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ∈ 𝐵)
185	1, 4, 63, 5, 3, 183, 184, 21	psrmulval 19386	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘𝑓 − 𝑘))))))
186	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑋 ∈ 𝐵)
187	14	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
188	1, 4, 63, 5, 3, 186, 187, 21	psrmulval 19386	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘𝑓 − 𝑗))))))
189	182, 185, 188	3eqtr4d 2666	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
190	13, 17, 189	eqfnfvd 6314	1 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ∘_𝑟 cofr 6896   supp csupp 7295   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275  ℂcc 9934   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  0_gc0g 16100   Σ_g cgsu 16101  CMndccmn 18193  Ringcrg 18547   mPwSer cmps 19351

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-ofr 6898  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  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-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-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-tset 15960  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-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-psr 19356

This theorem is referenced by:  psrring  19411