MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psrass1 Structured version   Visualization version   GIF version

Theorem psrass1 21984
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 2737 . . . 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 21966 . . . . 5 (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10 psrass.z . . . . 5 (𝜑𝑍𝐵)
111, 4, 5, 6, 9, 10psrmulcl 21966 . . . 4 (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
121, 2, 3, 4, 11psrelbas 21954 . . 3 (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
1312ffnd 6737 . 2 (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
141, 4, 5, 6, 8, 10psrmulcl 21966 . . . . 5 (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
151, 4, 5, 6, 7, 14psrmulcl 21966 . . . 4 (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
161, 2, 3, 4, 15psrelbas 21954 . . 3 (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
1716ffnd 6737 . 2 (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18 eqid 2737 . . . . 5 {𝑔𝐷𝑔r𝑥} = {𝑔𝐷𝑔r𝑥}
19 simpr 484 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝐷)
206ringcmnd 20281 . . . . . 6 (𝜑𝑅 ∈ CMnd)
2120adantr 480 . . . . 5 ((𝜑𝑥𝐷) → 𝑅 ∈ CMnd)
22 eqid 2737 . . . . . . 7 (.r𝑅) = (.r𝑅)
236ad3antrrr 730 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑅 ∈ Ring)
241, 2, 3, 4, 7psrelbas 21954 . . . . . . . . . 10 (𝜑𝑋:𝐷⟶(Base‘𝑅))
2524ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
26 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗 ∈ {𝑔𝐷𝑔r𝑥})
27 breq1 5146 . . . . . . . . . . . 12 (𝑔 = 𝑗 → (𝑔r𝑥𝑗r𝑥))
2827elrab 3692 . . . . . . . . . . 11 (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑗𝐷𝑗r𝑥))
2926, 28sylib 218 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗𝐷𝑗r𝑥))
3029simpld 494 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗𝐷)
3125, 30ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑋𝑗) ∈ (Base‘𝑅))
3231adantr 480 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑋𝑗) ∈ (Base‘𝑅))
331, 2, 3, 4, 8psrelbas 21954 . . . . . . . . . 10 (𝜑𝑌:𝐷⟶(Base‘𝑅))
3433ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
35 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})
36 breq1 5146 . . . . . . . . . . . 12 ( = 𝑛 → (r ≤ (𝑥f𝑗) ↔ 𝑛r ≤ (𝑥f𝑗)))
3736elrab 3692 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↔ (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
3835, 37sylib 218 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
3938simpld 494 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛𝐷)
4034, 39ffvelcdmd 7105 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑌𝑛) ∈ (Base‘𝑅))
411, 2, 3, 4, 10psrelbas 21954 . . . . . . . . . 10 (𝜑𝑍:𝐷⟶(Base‘𝑅))
4241ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
43 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
443psrbagf 21938 . . . . . . . . . . . . . . 15 (𝑗𝐷𝑗:𝐼⟶ℕ0)
4530, 44syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗:𝐼⟶ℕ0)
4629simprd 495 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗r𝑥)
473psrbagcon 21945 . . . . . . . . . . . . . 14 ((𝑥𝐷𝑗:𝐼⟶ℕ0𝑗r𝑥) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
4843, 45, 46, 47syl3anc 1373 . . . . . . . . . . . . 13 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
4948simpld 494 . . . . . . . . . . . 12 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑗) ∈ 𝐷)
5049adantr 480 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑥f𝑗) ∈ 𝐷)
513psrbagf 21938 . . . . . . . . . . . 12 (𝑛𝐷𝑛:𝐼⟶ℕ0)
5239, 51syl 17 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛:𝐼⟶ℕ0)
5338simprd 495 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛r ≤ (𝑥f𝑗))
543psrbagcon 21945 . . . . . . . . . . 11 (((𝑥f𝑗) ∈ 𝐷𝑛:𝐼⟶ℕ0𝑛r ≤ (𝑥f𝑗)) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5550, 52, 53, 54syl3anc 1373 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5655simpld 494 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑥f𝑗) ∘f𝑛) ∈ 𝐷)
5742, 56ffvelcdmd 7105 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅))
582, 22, 23, 40, 57ringcld 20257 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
592, 22, 23, 32, 58ringcld 20257 . . . . . 6 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
6059anasss 466 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
61 fveq2 6906 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑌𝑛) = (𝑌‘(𝑘f𝑗)))
62 oveq2 7439 . . . . . . . 8 (𝑛 = (𝑘f𝑗) → ((𝑥f𝑗) ∘f𝑛) = ((𝑥f𝑗) ∘f − (𝑘f𝑗)))
6362fveq2d 6910 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) = (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))
6461, 63oveq12d 7449 . . . . . 6 (𝑛 = (𝑘f𝑗) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))
6564oveq2d 7447 . . . . 5 (𝑛 = (𝑘f𝑗) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
663, 18, 19, 2, 21, 60, 65psrass1lem 21952 . . . 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)))))))))
677ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋𝐵)
688ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
69 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘 ∈ {𝑔𝐷𝑔r𝑥})
70 breq1 5146 . . . . . . . . . . . 12 (𝑔 = 𝑘 → (𝑔r𝑥𝑘r𝑥))
7170elrab 3692 . . . . . . . . . . 11 (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑘𝐷𝑘r𝑥))
7269, 71sylib 218 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑘𝐷𝑘r𝑥))
7372simpld 494 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘𝐷)
741, 4, 22, 5, 3, 67, 68, 73psrmulval 21964 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))))
7574oveq1d 7446 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
76 eqid 2737 . . . . . . . 8 (0g𝑅) = (0g𝑅)
776ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
783psrbaglefi 21946 . . . . . . . . 9 (𝑘𝐷 → {𝐷r𝑘} ∈ Fin)
7973, 78syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r𝑘} ∈ Fin)
8041ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
81 simplr 769 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
823psrbagf 21938 . . . . . . . . . . . 12 (𝑘𝐷𝑘:𝐼⟶ℕ0)
8373, 82syl 17 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘:𝐼⟶ℕ0)
8472simprd 495 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘r𝑥)
853psrbagcon 21945 . . . . . . . . . . 11 ((𝑥𝐷𝑘:𝐼⟶ℕ0𝑘r𝑥) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
8681, 83, 84, 85syl3anc 1373 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
8786simpld 494 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑘) ∈ 𝐷)
8880, 87ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
896ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑅 ∈ Ring)
9024ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
91 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 ∈ {𝐷r𝑘})
92 breq1 5146 . . . . . . . . . . . . 13 ( = 𝑗 → (r𝑘𝑗r𝑘))
9392elrab 3692 . . . . . . . . . . . 12 (𝑗 ∈ {𝐷r𝑘} ↔ (𝑗𝐷𝑗r𝑘))
9491, 93sylib 218 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑗𝐷𝑗r𝑘))
9594simpld 494 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗𝐷)
9690, 95ffvelcdmd 7105 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑋𝑗) ∈ (Base‘𝑅))
9733ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
9873adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘𝐷)
9995, 44syl 17 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗:𝐼⟶ℕ0)
10094simprd 495 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗r𝑘)
1013psrbagcon 21945 . . . . . . . . . . . 12 ((𝑘𝐷𝑗:𝐼⟶ℕ0𝑗r𝑘) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
10298, 99, 100, 101syl3anc 1373 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
103102simpld 494 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
10497, 103ffvelcdmd 7105 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅))
1052, 22, 89, 96, 104ringcld 20257 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
106 eqid 2737 . . . . . . . . 9 (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))
107 fvex 6919 . . . . . . . . . 10 (0g𝑅) ∈ V
108107a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (0g𝑅) ∈ V)
109106, 79, 105, 108fsuppmptdm 9416 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) finSupp (0g𝑅))
1102, 76, 22, 77, 79, 88, 105, 109gsummulc1 20313 . . . . . . 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))))
11188adantr 480 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
1122, 22ringass 20250 . . . . . . . . . . 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)))))
11389, 96, 104, 111, 112syl13anc 1374 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
1143psrbagf 21938 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷𝑥:𝐼⟶ℕ0)
115114ad3antlr 731 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥:𝐼⟶ℕ0)
116115ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑥𝑧) ∈ ℕ0)
11783adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘:𝐼⟶ℕ0)
118117ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
11999ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
120 nn0cn 12536 . . . . . . . . . . . . . . . . 17 ((𝑥𝑧) ∈ ℕ0 → (𝑥𝑧) ∈ ℂ)
121 nn0cn 12536 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
122 nn0cn 12536 . . . . . . . . . . . . . . . . 17 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
123 nnncan2 11546 . . . . . . . . . . . . . . . . 17 (((𝑥𝑧) ∈ ℂ ∧ (𝑘𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
124120, 121, 122, 123syl3an 1161 . . . . . . . . . . . . . . . 16 (((𝑥𝑧) ∈ ℕ0 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
125116, 118, 119, 124syl3anc 1373 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
126125mpteq2dva 5242 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
127 psrring.i . . . . . . . . . . . . . . . 16 (𝜑𝐼𝑉)
128127ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝐼𝑉)
129 ovexd 7466 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑥𝑧) − (𝑗𝑧)) ∈ V)
130 ovexd 7466 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑘𝑧) − (𝑗𝑧)) ∈ V)
131115feqmptd 6977 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥 = (𝑧𝐼 ↦ (𝑥𝑧)))
13299feqmptd 6977 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
133128, 116, 119, 131, 132offval2 7717 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑗) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑗𝑧))))
134117feqmptd 6977 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
135128, 118, 119, 134, 132offval2 7717 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑗𝑧))))
136128, 129, 130, 133, 135offval2 7717 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))))
137128, 116, 118, 131, 134offval2 7717 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑘) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
138126, 136, 1373eqtr4d 2787 . . . . . . . . . . . . 13 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑥f𝑘))
139138fveq2d 6910 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))) = (𝑍‘(𝑥f𝑘)))
140139oveq2d 7447 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘))))
141140oveq2d 7447 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
142113, 141eqtr4d 2780 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
143142mpteq2dva 5242 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))
144143oveq2d 7447 . . . . . . 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))))))))
14575, 110, 1443eqtr2d 2783 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
146145mpteq2dva 5242 . . . . 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)))))))))
147146oveq2d 7447 . . . 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))))))))))
1488ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
14910ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍𝐵)
1501, 4, 22, 5, 3, 148, 149, 49psrmulval 21964 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑌 × 𝑍)‘(𝑥f𝑗)) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))
151150oveq2d 7447 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
1526ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
1533psrbaglefi 21946 . . . . . . . . 9 ((𝑥f𝑗) ∈ 𝐷 → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
15449, 153syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
155 ovex 7464 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
1563, 155rab2ex 5342 . . . . . . . . . . . 12 {𝐷r ≤ (𝑥f𝑗)} ∈ V
157156mptex 7243 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V
158 funmpt 6604 . . . . . . . . . . 11 Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
159157, 158, 1073pm3.2i 1340 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V)
160159a1i 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))
161 suppssdm 8202 . . . . . . . . . . 11 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
162 eqid 2737 . . . . . . . . . . . 12 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
163162dmmptss 6261 . . . . . . . . . . 11 dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ⊆ {𝐷r ≤ (𝑥f𝑗)}
164161, 163sstri 3993 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)}
165164a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})
166 suppssfifsupp 9420 . . . . . . . . 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))
167160, 154, 165, 166syl12anc 837 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
1682, 76, 22, 152, 154, 31, 58, 167gsummulc2 20314 . . . . . . 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)))))))
169151, 168eqtr4d 2780 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
170169mpteq2dva 5242 . . . . 5 ((𝜑𝑥𝐷) → (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗)))) = (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))))
171170oveq2d 7447 . . . 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)))))))))
17266, 147, 1713eqtr4d 2787 . . 3 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
1739adantr 480 . . . 4 ((𝜑𝑥𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
17410adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝑍𝐵)
1751, 4, 22, 5, 3, 173, 174, 19psrmulval 21964 . . 3 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))))
1767adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝑋𝐵)
17714adantr 480 . . . 4 ((𝜑𝑥𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
1781, 4, 22, 5, 3, 176, 177, 19psrmulval 21964 . . 3 ((𝜑𝑥𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
179172, 175, 1783eqtr4d 2787 . 2 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
18013, 17, 179eqfnfvd 7054 1 (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  ccnv 5684  dom cdm 5685  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  r cofr 7696   supp csupp 8185  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  cc 11153  cle 11296  cmin 11492  cn 12266  0cn0 12526  Basecbs 17247  .rcmulr 17298  0gc0g 17484   Σg cgsu 17485  CMndccmn 19798  Ringcrg 20230   mPwSer cmps 21924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-mulg 19086  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-ur 20179  df-ring 20232  df-psr 21929
This theorem is referenced by:  psrring  21990
  Copyright terms: Public domain W3C validator