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Theorem psrass1 20650
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 (𝜑𝐼𝑉)
psrring.r (𝜑𝑅 ∈ Ring)
psrass.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrass.t × = (.r𝑆)
psrass.b 𝐵 = (Base‘𝑆)
psrass.x (𝜑𝑋𝐵)
psrass.y (𝜑𝑌𝐵)
psrass.z (𝜑𝑍𝐵)
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 𝑗 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrring.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
2 eqid 2824 . . . 4 (Base‘𝑅) = (Base‘𝑅)
3 psrass.d . . . 4 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
4 psrass.b . . . 4 𝐵 = (Base‘𝑆)
5 psrass.t . . . . 5 × = (.r𝑆)
6 psrring.r . . . . 5 (𝜑𝑅 ∈ Ring)
7 psrass.x . . . . . 6 (𝜑𝑋𝐵)
8 psrass.y . . . . . 6 (𝜑𝑌𝐵)
91, 4, 5, 6, 7, 8psrmulcl 20633 . . . . 5 (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10 psrass.z . . . . 5 (𝜑𝑍𝐵)
111, 4, 5, 6, 9, 10psrmulcl 20633 . . . 4 (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
121, 2, 3, 4, 11psrelbas 20624 . . 3 (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
1312ffnd 6504 . 2 (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
141, 4, 5, 6, 8, 10psrmulcl 20633 . . . . 5 (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
151, 4, 5, 6, 7, 14psrmulcl 20633 . . . 4 (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
161, 2, 3, 4, 15psrelbas 20624 . . 3 (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
1716ffnd 6504 . 2 (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18 eqid 2824 . . . . 5 {𝑔𝐷𝑔r𝑥} = {𝑔𝐷𝑔r𝑥}
19 psrring.i . . . . . 6 (𝜑𝐼𝑉)
2019adantr 484 . . . . 5 ((𝜑𝑥𝐷) → 𝐼𝑉)
21 simpr 488 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝐷)
22 ringcmn 19334 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
236, 22syl 17 . . . . . 6 (𝜑𝑅 ∈ CMnd)
2423adantr 484 . . . . 5 ((𝜑𝑥𝐷) → 𝑅 ∈ CMnd)
256ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
2625adantr 484 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑅 ∈ Ring)
271, 2, 3, 4, 7psrelbas 20624 . . . . . . . . . 10 (𝜑𝑋:𝐷⟶(Base‘𝑅))
2827ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
29 simpr 488 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗 ∈ {𝑔𝐷𝑔r𝑥})
30 breq1 5055 . . . . . . . . . . . 12 (𝑔 = 𝑗 → (𝑔r𝑥𝑗r𝑥))
3130elrab 3666 . . . . . . . . . . 11 (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑗𝐷𝑗r𝑥))
3229, 31sylib 221 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗𝐷𝑗r𝑥))
3332simpld 498 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗𝐷)
3428, 33ffvelrnd 6843 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑋𝑗) ∈ (Base‘𝑅))
3534adantr 484 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑋𝑗) ∈ (Base‘𝑅))
361, 2, 3, 4, 8psrelbas 20624 . . . . . . . . . 10 (𝜑𝑌:𝐷⟶(Base‘𝑅))
3736ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
38 simpr 488 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})
39 breq1 5055 . . . . . . . . . . . 12 ( = 𝑛 → (r ≤ (𝑥f𝑗) ↔ 𝑛r ≤ (𝑥f𝑗)))
4039elrab 3666 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↔ (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
4138, 40sylib 221 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
4241simpld 498 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛𝐷)
4337, 42ffvelrnd 6843 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑌𝑛) ∈ (Base‘𝑅))
441, 2, 3, 4, 10psrelbas 20624 . . . . . . . . . 10 (𝜑𝑍:𝐷⟶(Base‘𝑅))
4544ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
4619ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝐼𝑉)
4746adantr 484 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝐼𝑉)
48 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
493psrbagf 20610 . . . . . . . . . . . . . . 15 ((𝐼𝑉𝑗𝐷) → 𝑗:𝐼⟶ℕ0)
5046, 33, 49syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗:𝐼⟶ℕ0)
5132simprd 499 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗r𝑥)
523psrbagcon 20616 . . . . . . . . . . . . . 14 ((𝐼𝑉 ∧ (𝑥𝐷𝑗:𝐼⟶ℕ0𝑗r𝑥)) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
5346, 48, 50, 51, 52syl13anc 1369 . . . . . . . . . . . . 13 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
5453simpld 498 . . . . . . . . . . . 12 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑗) ∈ 𝐷)
5554adantr 484 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑥f𝑗) ∈ 𝐷)
563psrbagf 20610 . . . . . . . . . . . 12 ((𝐼𝑉𝑛𝐷) → 𝑛:𝐼⟶ℕ0)
5747, 42, 56syl2anc 587 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛:𝐼⟶ℕ0)
5841simprd 499 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛r ≤ (𝑥f𝑗))
593psrbagcon 20616 . . . . . . . . . . 11 ((𝐼𝑉 ∧ ((𝑥f𝑗) ∈ 𝐷𝑛:𝐼⟶ℕ0𝑛r ≤ (𝑥f𝑗))) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
6047, 55, 57, 58, 59syl13anc 1369 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
6160simpld 498 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑥f𝑗) ∘f𝑛) ∈ 𝐷)
6245, 61ffvelrnd 6843 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅))
63 eqid 2824 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
642, 63ringcl 19314 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑌𝑛) ∈ (Base‘𝑅) ∧ (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅)) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
6526, 43, 62, 64syl3anc 1368 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
662, 63ringcl 19314 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑋𝑗) ∈ (Base‘𝑅) ∧ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅)) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
6726, 35, 65, 66syl3anc 1368 . . . . . 6 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
6867anasss 470 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
69 fveq2 6661 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑌𝑛) = (𝑌‘(𝑘f𝑗)))
70 oveq2 7157 . . . . . . . 8 (𝑛 = (𝑘f𝑗) → ((𝑥f𝑗) ∘f𝑛) = ((𝑥f𝑗) ∘f − (𝑘f𝑗)))
7170fveq2d 6665 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) = (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))
7269, 71oveq12d 7167 . . . . . 6 (𝑛 = (𝑘f𝑗) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))
7372oveq2d 7165 . . . . 5 (𝑛 = (𝑘f𝑗) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
743, 18, 20, 21, 2, 24, 68, 73psrass1lem 20622 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))))
757ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋𝐵)
768ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
77 simpr 488 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘 ∈ {𝑔𝐷𝑔r𝑥})
78 breq1 5055 . . . . . . . . . . . 12 (𝑔 = 𝑘 → (𝑔r𝑥𝑘r𝑥))
7978elrab 3666 . . . . . . . . . . 11 (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑘𝐷𝑘r𝑥))
8077, 79sylib 221 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑘𝐷𝑘r𝑥))
8180simpld 498 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘𝐷)
821, 4, 63, 5, 3, 75, 76, 81psrmulval 20631 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))))
8382oveq1d 7164 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
84 eqid 2824 . . . . . . . 8 (0g𝑅) = (0g𝑅)
85 eqid 2824 . . . . . . . 8 (+g𝑅) = (+g𝑅)
866ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
8719ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝐼𝑉)
883psrbaglefi 20617 . . . . . . . . 9 ((𝐼𝑉𝑘𝐷) → {𝐷r𝑘} ∈ Fin)
8987, 81, 88syl2anc 587 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r𝑘} ∈ Fin)
9044ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
91 simplr 768 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
923psrbagf 20610 . . . . . . . . . . . 12 ((𝐼𝑉𝑘𝐷) → 𝑘:𝐼⟶ℕ0)
9387, 81, 92syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘:𝐼⟶ℕ0)
9480simprd 499 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘r𝑥)
953psrbagcon 20616 . . . . . . . . . . 11 ((𝐼𝑉 ∧ (𝑥𝐷𝑘:𝐼⟶ℕ0𝑘r𝑥)) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
9687, 91, 93, 94, 95syl13anc 1369 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
9796simpld 498 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑘) ∈ 𝐷)
9890, 97ffvelrnd 6843 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
9986adantr 484 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑅 ∈ Ring)
10027ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
101 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 ∈ {𝐷r𝑘})
102 breq1 5055 . . . . . . . . . . . . 13 ( = 𝑗 → (r𝑘𝑗r𝑘))
103102elrab 3666 . . . . . . . . . . . 12 (𝑗 ∈ {𝐷r𝑘} ↔ (𝑗𝐷𝑗r𝑘))
104101, 103sylib 221 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑗𝐷𝑗r𝑘))
105104simpld 498 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗𝐷)
106100, 105ffvelrnd 6843 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑋𝑗) ∈ (Base‘𝑅))
10736ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
10887adantr 484 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝐼𝑉)
10981adantr 484 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘𝐷)
110108, 105, 49syl2anc 587 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗:𝐼⟶ℕ0)
111104simprd 499 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗r𝑘)
1123psrbagcon 20616 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ (𝑘𝐷𝑗:𝐼⟶ℕ0𝑗r𝑘)) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
113108, 109, 110, 111, 112syl13anc 1369 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
114113simpld 498 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
115107, 114ffvelrnd 6843 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅))
1162, 63ringcl 19314 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑋𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅)) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
11799, 106, 115, 116syl3anc 1368 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
118 eqid 2824 . . . . . . . . 9 (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))
119 fvex 6674 . . . . . . . . . 10 (0g𝑅) ∈ V
120119a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (0g𝑅) ∈ V)
121118, 89, 117, 120fsuppmptdm 8841 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) finSupp (0g𝑅))
1222, 84, 85, 63, 86, 89, 98, 117, 121gsummulc1 19359 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
12398adantr 484 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
1242, 63ringass 19317 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((𝑋𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
12599, 106, 115, 123, 124syl13anc 1369 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
1263psrbagf 20610 . . . . . . . . . . . . . . . . . . 19 ((𝐼𝑉𝑥𝐷) → 𝑥:𝐼⟶ℕ0)
12719, 126sylan 583 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐷) → 𝑥:𝐼⟶ℕ0)
128127ad2antrr 725 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥:𝐼⟶ℕ0)
129128ffvelrnda 6842 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑥𝑧) ∈ ℕ0)
13093adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘:𝐼⟶ℕ0)
131130ffvelrnda 6842 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
132110ffvelrnda 6842 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
133 nn0cn 11904 . . . . . . . . . . . . . . . . 17 ((𝑥𝑧) ∈ ℕ0 → (𝑥𝑧) ∈ ℂ)
134 nn0cn 11904 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
135 nn0cn 11904 . . . . . . . . . . . . . . . . 17 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
136 nnncan2 10921 . . . . . . . . . . . . . . . . 17 (((𝑥𝑧) ∈ ℂ ∧ (𝑘𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
137133, 134, 135, 136syl3an 1157 . . . . . . . . . . . . . . . 16 (((𝑥𝑧) ∈ ℕ0 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
138129, 131, 132, 137syl3anc 1368 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
139138mpteq2dva 5147 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
140 ovexd 7184 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑥𝑧) − (𝑗𝑧)) ∈ V)
141 ovexd 7184 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑘𝑧) − (𝑗𝑧)) ∈ V)
142128feqmptd 6724 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥 = (𝑧𝐼 ↦ (𝑥𝑧)))
143110feqmptd 6724 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
144108, 129, 132, 142, 143offval2 7420 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑗) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑗𝑧))))
145130feqmptd 6724 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
146108, 131, 132, 145, 143offval2 7420 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑗𝑧))))
147108, 140, 141, 144, 146offval2 7420 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))))
148108, 129, 131, 142, 145offval2 7420 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑘) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
149139, 147, 1483eqtr4d 2869 . . . . . . . . . . . . 13 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑥f𝑘))
150149fveq2d 6665 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))) = (𝑍‘(𝑥f𝑘)))
151150oveq2d 7165 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘))))
152151oveq2d 7165 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
153125, 152eqtr4d 2862 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
154153mpteq2dva 5147 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))
155154oveq2d 7165 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
15683, 122, 1553eqtr2d 2865 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
157156mpteq2dva 5147 . . . . 5 ((𝜑𝑥𝐷) → (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))))
158157oveq2d 7165 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))))
1598ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
16010ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍𝐵)
1611, 4, 63, 5, 3, 159, 160, 54psrmulval 20631 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑌 × 𝑍)‘(𝑥f𝑗)) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))
162161oveq2d 7165 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
1633psrbaglefi 20617 . . . . . . . . 9 ((𝐼𝑉 ∧ (𝑥f𝑗) ∈ 𝐷) → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
16446, 54, 163syl2anc 587 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
165 ovex 7182 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
1663, 165rab2ex 5224 . . . . . . . . . . . 12 {𝐷r ≤ (𝑥f𝑗)} ∈ V
167166mptex 6977 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V
168 funmpt 6381 . . . . . . . . . . 11 Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
169167, 168, 1193pm3.2i 1336 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V)
170169a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V))
171 suppssdm 7839 . . . . . . . . . . 11 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
172 eqid 2824 . . . . . . . . . . . 12 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
173172dmmptss 6082 . . . . . . . . . . 11 dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ⊆ {𝐷r ≤ (𝑥f𝑗)}
174171, 173sstri 3962 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)}
175174a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})
176 suppssfifsupp 8845 . . . . . . . . 9 ((((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V) ∧ ({𝐷r ≤ (𝑥f𝑗)} ∈ Fin ∧ ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
177170, 164, 175, 176syl12anc 835 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
1782, 84, 85, 63, 25, 164, 34, 65, 177gsummulc2 19360 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
179162, 178eqtr4d 2862 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
180179mpteq2dva 5147 . . . . 5 ((𝜑𝑥𝐷) → (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗)))) = (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))))
181180oveq2d 7165 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))))
18274, 158, 1813eqtr4d 2869 . . 3 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
1839adantr 484 . . . 4 ((𝜑𝑥𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
18410adantr 484 . . . 4 ((𝜑𝑥𝐷) → 𝑍𝐵)
1851, 4, 63, 5, 3, 183, 184, 21psrmulval 20631 . . 3 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))))
1867adantr 484 . . . 4 ((𝜑𝑥𝐷) → 𝑋𝐵)
18714adantr 484 . . . 4 ((𝜑𝑥𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
1881, 4, 63, 5, 3, 186, 187, 21psrmulval 20631 . . 3 ((𝜑𝑥𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
189182, 185, 1883eqtr4d 2869 . 2 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
19013, 17, 189eqfnfvd 6796 1 (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  wss 3919   class class class wbr 5052  cmpt 5132  ccnv 5541  dom cdm 5542  cima 5545  Fun wfun 6337  wf 6339  cfv 6343  (class class class)co 7149  f cof 7401  r cofr 7402   supp csupp 7826  m cmap 8402  Fincfn 8505   finSupp cfsupp 8830  cc 10533  cle 10674  cmin 10868  cn 11634  0cn0 11894  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  0gc0g 16713   Σg cgsu 16714  CMndccmn 18906  Ringcrg 19297   mPwSer cmps 20596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-mulg 18225  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-psr 20601
This theorem is referenced by:  psrring  20656
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