Theorem psrass1 19226

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 2610	. . . 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 19209	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10		psrass.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	1, 4, 5, 6, 9, 10	psrmulcl 19209	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
12	1, 2, 3, 4, 11	psrelbas 19200	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
13	12	ffnd 5959	. 2 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
14	1, 4, 5, 6, 8, 10	psrmulcl 19209	. . . . 5 ⊢ (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
15	1, 4, 5, 6, 7, 14	psrmulcl 19209	. . . 4 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
16	1, 2, 3, 4, 15	psrelbas 19200	. . 3 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
17	16	ffnd 5959	. 2 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18		eqid 2610	. . . . 5 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}
19		psrring.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐼 ∈ 𝑉)
21		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
22		ringcmn 18404	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
23	6, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd)
25	6	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑅 ∈ Ring)
26	25	adantr 480	. . . . . . 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 19200	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
28	27	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
29		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
30		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑗 ∘𝑟 ≤ 𝑥))
31	30	elrab 3331	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑥))
32	29, 31	sylib 207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑥))
33	32	simpld 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ 𝐷)
34	28, 33	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
35	34	adantr 480	. . . . . . 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 19200	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
37	36	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
38		simpr 476	. . . . . . . . . . 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 4586	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗) ↔ 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
40	39	elrab 3331	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↔ (𝑛 ∈ 𝐷 ∧ 𝑛 ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)))
41	38, 40	sylib 207	. . . . . . . . . 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 474	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑛 ∈ 𝐷)
43	37, 42	ffvelrnd 6268	. . . . . . . 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 19200	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:𝐷⟶(Base‘𝑅))
45	44	ad3antrrr 762	. . . . . . . . 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 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝐼 ∈ 𝑉)
47	46	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝐼 ∈ 𝑉)
48		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
49	3	psrbagf 19186	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐷) → 𝑗:𝐼⟶ℕ0)
50	46, 33, 49	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
51	32	simprd 478	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∘𝑟 ≤ 𝑥)
52	3	psrbagcon 19192	. . . . . . . . . . . . . 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 1320	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑥))
54	53	simpld 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷)
55	54	adantr 480	. . . . . . . . . . 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 19186	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑛 ∈ 𝐷) → 𝑛:𝐼⟶ℕ0)
57	47, 42, 56	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)}) → 𝑛:𝐼⟶ℕ0)
58	41	simprd 478	. . . . . . . . . . 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 19192	. . . . . . . . . . 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 1320	. . . . . . . . . 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 474	. . . . . . . . 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 6268	. . . . . . . 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 2610	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
64	2, 63	ringcl 18384	. . . . . . . 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 1318	. . . . . . 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 18384	. . . . . . 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 1318	. . . . . 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 677	. . . . 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 6103	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → (𝑌‘𝑛) = (𝑌‘(𝑘 ∘𝑓 − 𝑗)))
70		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛) = ((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗)))
71	70	fveq2d 6107	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘𝑓 − 𝑗) → (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)) = (𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − (𝑘 ∘𝑓 − 𝑗))))
72	69, 71	oveq12d 6567	. . . . . 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 6565	. . . . 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 19198	. . . 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 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑋 ∈ 𝐵)
76	8	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑌 ∈ 𝐵)
77		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
78		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑘 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑘 ∘𝑟 ≤ 𝑥))
79	78	elrab 3331	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘𝑟 ≤ 𝑥))
80	77, 79	sylib 207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘𝑟 ≤ 𝑥))
81	80	simpld 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ 𝐷)
82	1, 4, 63, 5, 3, 75, 76, 81	psrmulval 19207	. . . . . . . 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 6564	. . . . . . 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 2610	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
85		eqid 2610	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
86	6	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑅 ∈ Ring)
87	19	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝐼 ∈ 𝑉)
88	3	psrbaglefi 19193	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ∈ Fin)
89	87, 81, 88	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ∈ Fin)
90	44	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
91		simplr 788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
92	3	psrbagf 19186	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0)
93	87, 81, 92	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
94	80	simprd 478	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∘𝑟 ≤ 𝑥)
95	3	psrbagcon 19192	. . . . . . . . . . 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 1320	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑘) ∘𝑟 ≤ 𝑥))
97	96	simpld 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑘) ∈ 𝐷)
98	90, 97	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑍‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅))
99	86	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring)
100	27	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
101		simpr 476	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘})
102		breq1 4586	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑗 → (ℎ ∘𝑟 ≤ 𝑘 ↔ 𝑗 ∘𝑟 ≤ 𝑘))
103	102	elrab 3331	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑘))
104	101, 103	sylib 207	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑘))
105	104	simpld 474	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ 𝐷)
106	100, 105	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
107	36	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
108	87	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑉)
109	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷)
110	108, 105, 49	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗:𝐼⟶ℕ0)
111	104	simprd 478	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 ∘𝑟 ≤ 𝑘)
112	3	psrbagcon 19192	. . . . . . . . . . . 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 1320	. . . . . . . . . . 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 474	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ 𝐷)
115	107, 114	ffvelrnd 6268	. . . . . . . . 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 18384	. . . . . . . . 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 1318	. . . . . . . 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 2610	. . . . . . . . 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 6113	. . . . . . . . . 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 8169	. . . . . . . 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 18429	. . . . . . 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 480	. . . . . . . . . . 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 18387	. . . . . . . . . . 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 1320	. . . . . . . . . 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 19186	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
127	19, 126	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
128	127	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0)
129	128	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
130	93	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
131	130	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
132	110	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
133		nn0cn 11179	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
134		nn0cn 11179	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
135		nn0cn 11179	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
136		nnncan2 10197	. . . . . . . . . . . . . . . . 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 1360	. . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . 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 4672	. . . . . . . . . . . . . 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		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V
141	140	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V)
142		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V
143	142	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V)
144	128	feqmptd 6159	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
145	110	feqmptd 6159	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
146	108, 129, 132, 144, 145	offval2 6812	. . . . . . . . . . . . . . 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))))
147	130	feqmptd 6159	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
148	108, 131, 132, 147, 145	offval2 6812	. . . . . . . . . . . . . . 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))))
149	108, 141, 143, 146, 148	offval2 6812	. . . . . . . . . . . . . 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)))))
150	108, 129, 131, 144, 147	offval2 6812	. . . . . . . . . . . . . 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))))
151	139, 149, 150	3eqtr4d 2654	. . . . . . . . . . . . 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))
152	151	fveq2d 6107	. . . . . . . . . . . 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)))
153	152	oveq2d 6565	. . . . . . . . . . 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))))
154	153	oveq2d 6565	. . . . . . . . . 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)))))
155	125, 154	eqtr4d 2647	. . . . . . . . 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))))))
156	155	mpteq2dva 4672	. . . . . . . 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)))))))
157	156	oveq2d 6565	. . . . . . 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))))))))
158	83, 122, 157	3eqtr2d 2650	. . . . . 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))))))))
159	158	mpteq2dva 4672	. . . . 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)))))))))
160	159	oveq2d 6565	. . . 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))))))))))
161	8	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑌 ∈ 𝐵)
162	10	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑍 ∈ 𝐵)
163	1, 4, 63, 5, 3, 161, 162, 54	psrmulval 19207	. . . . . . . 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))))))
164	163	oveq2d 6565	. . . . . . 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)))))))
165	3	psrbaglefi 19193	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ Fin)
166	46, 54, 165	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ Fin)
167		ovex 6577	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
168	3, 167	rab2ex 4743	. . . . . . . . . . . 12 ⊢ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ∈ V
169	168	mptex 6390	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛)))) ∈ V
170		funmpt 5840	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘𝑟 ≤ (𝑥 ∘𝑓 − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘𝑓 − 𝑗) ∘𝑓 − 𝑛))))
171	169, 170, 119	3pm3.2i 1232	. . . . . . . . . 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)
172	171	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))
173		suppssdm 7195	. . . . . . . . . . 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))))
174		eqid 2610	. . . . . . . . . . . 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))))
175	174	dmmptss 5548	. . . . . . . . . . 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)}
176	173, 175	sstri 3577	. . . . . . . . . 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)}
177	176	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)})
178		suppssfifsupp 8173	. . . . . . . . 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))
179	172, 166, 177, 178	syl12anc 1316	. . . . . . . 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))
180	2, 84, 85, 63, 25, 166, 34, 65, 179	gsummulc2 18430	. . . . . . 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)))))))
181	164, 180	eqtr4d 2647	. . . . . 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)))))))
182	181	mpteq2dva 4672	. . . . 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))))))))
183	182	oveq2d 6565	. . . 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)))))))))
184	74, 160, 183	3eqtr4d 2654	. . 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))))))
185	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
186	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ∈ 𝐵)
187	1, 4, 63, 5, 3, 185, 186, 21	psrmulval 19207	. . 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))))))
188	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑋 ∈ 𝐵)
189	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
190	1, 4, 63, 5, 3, 188, 189, 21	psrmulval 19207	. . 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))))))
191	184, 187, 190	3eqtr4d 2654	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
192	13, 17, 191	eqfnfvd 6222	1 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ∘_𝑟 cofr 6794   supp csupp 7182   ↑_𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  ℂcc 9813   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  0_gc0g 15923   Σ_g cgsu 15924  CMndccmn 18016  Ringcrg 18370   mPwSer cmps 19172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-psr 19177

This theorem is referenced by:  psrring  19232