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Theorem psrval 21270
Description: Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
psrval.s 𝑆 = (𝐌 mPwSer 𝑅)
psrval.k 𝐟 = (Base‘𝑅)
psrval.a + = (+g‘𝑅)
psrval.m · = (.r‘𝑅)
psrval.o 𝑂 = (TopOpen‘𝑅)
psrval.d 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐌) ∣ (◡ℎ “ ℕ) ∈ Fin}
psrval.b (𝜑 → 𝐵 = (𝐟 ↑m 𝐷))
psrval.p ✚ = ( ∘f + ↟ (𝐵 × 𝐵))
psrval.t × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↩ (𝑘 ∈ 𝐷 ↩ (𝑅 Σg (𝑥 ∈ {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥)))))))
psrval.v ∙ = (𝑥 ∈ 𝐟, 𝑓 ∈ 𝐵 ↩ ((𝐷 × {𝑥}) ∘f · 𝑓))
psrval.j (𝜑 → 𝐜 = (∏t‘(𝐷 × {𝑂})))
psrval.i (𝜑 → 𝐌 ∈ 𝑊)
psrval.r (𝜑 → 𝑅 ∈ 𝑋)
Assertion
Ref Expression
psrval (𝜑 → 𝑆 = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
Distinct variable groups:   𝑊,ℎ   𝑓,𝑔,𝑘,𝑥,𝜑   𝐵,𝑓,𝑔,𝑘,𝑥   𝑓,ℎ,𝐌,𝑔,𝑘,𝑥   𝑅,𝑓,𝑔,𝑘,𝑥   𝑊,𝑓,𝐷,𝑔,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑊,ℎ)   𝐵(𝑊,ℎ)   𝐷(ℎ)   + (𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   ✚ (𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   𝑅(𝑊,ℎ)   𝑆(𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   ∙ (𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   · (𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   × (𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   𝐌(𝑊)   𝐜(𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   𝐟(𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   𝑂(𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   𝑊(𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)   𝑋(𝑥,𝑊,𝑓,𝑔,ℎ,𝑘)

Proof of Theorem psrval
Dummy variables 𝑖 𝑟 𝑏 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrval.s . 2 𝑆 = (𝐌 mPwSer 𝑅)
2 df-psr 21264 . . . 4 mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↩ ⩋{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⊌⊋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
32a1i 11 . . 3 (𝜑 → mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↩ ⩋{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⊌⊋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩})))
4 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → 𝑖 = 𝐌)
54oveq2d 7367 . . . . . . 7 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐌))
6 rabeq 3419 . . . . . . 7 ((ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐌) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐌) ∣ (◡ℎ “ ℕ) ∈ Fin})
75, 6syl 17 . . . . . 6 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐌) ∣ (◡ℎ “ ℕ) ∈ Fin})
8 psrval.d . . . . . 6 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐌) ∣ (◡ℎ “ ℕ) ∈ Fin}
97, 8eqtr4di 2795 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
109csbeq1d 3857 . . . 4 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → ⩋{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⊌⊋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ⊋𝐷 / 𝑑⊌⊋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
11 ovex 7384 . . . . . . 7 (ℕ0 ↑m 𝑖) ∈ V
1211rabex 5287 . . . . . 6 {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
139, 12eqeltrrdi 2847 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → 𝐷 ∈ V)
14 simplrr 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑟 = 𝑅)
1514fveq2d 6843 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = (Base‘𝑅))
16 psrval.k . . . . . . . . . 10 𝐟 = (Base‘𝑅)
1715, 16eqtr4di 2795 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = 𝐟)
18 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1917, 18oveq12d 7369 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = (𝐟 ↑m 𝐷))
20 psrval.b . . . . . . . . 9 (𝜑 → 𝐵 = (𝐟 ↑m 𝐷))
2120ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 = (𝐟 ↑m 𝐷))
2219, 21eqtr4d 2780 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = 𝐵)
2322csbeq1d 3857 . . . . . 6 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ⩋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ⊋𝐵 / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
24 ovex 7384 . . . . . . . 8 ((Base‘𝑟) ↑m 𝑑) ∈ V
2522, 24eqeltrrdi 2847 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 ∈ V)
26 simpr 485 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2726opeq2d 4835 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟹(Base‘ndx), 𝑏⟩ = ⟹(Base‘ndx), 𝐵⟩)
2814adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
2928fveq2d 6843 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g‘𝑟) = (+g‘𝑅))
30 psrval.a . . . . . . . . . . . . . 14 + = (+g‘𝑅)
3129, 30eqtr4di 2795 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g‘𝑟) = + )
3231ofeqd 7611 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (+g‘𝑟) = ∘f + )
3326, 26xpeq12d 5662 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
3432, 33reseq12d 5936 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏)) = ( ∘f + ↟ (𝐵 × 𝐵)))
35 psrval.p . . . . . . . . . . 11 ✚ = ( ∘f + ↟ (𝐵 × 𝐵))
3634, 35eqtr4di 2795 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏)) = ✚ )
3736opeq2d 4835 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩ = ⟹(+g‘ndx), ✚ ⟩)
3818adantr 481 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
39 rabeq 3419 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} = {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘})
4038, 39syl 17 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} = {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘})
4128fveq2d 6843 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r‘𝑟) = (.r‘𝑅))
42 psrval.m . . . . . . . . . . . . . . . . 17 · = (.r‘𝑅)
4341, 42eqtr4di 2795 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r‘𝑟) = · )
4443oveqd 7368 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥))) = ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥))))
4540, 44mpteq12dv 5194 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))) = (𝑥 ∈ {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥)))))
4628, 45oveq12d 7369 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥))))) = (𝑅 Σg (𝑥 ∈ {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥))))))
4738, 46mpteq12dv 5194 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))) = (𝑘 ∈ 𝐷 ↩ (𝑅 Σg (𝑥 ∈ {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥)))))))
4826, 26, 47mpoeq123dv 7426 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥))))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↩ (𝑘 ∈ 𝐷 ↩ (𝑅 Σg (𝑥 ∈ {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥))))))))
49 psrval.t . . . . . . . . . . 11 × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↩ (𝑘 ∈ 𝐷 ↩ (𝑅 Σg (𝑥 ∈ {𝑊 ∈ 𝐷 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥)))))))
5048, 49eqtr4di 2795 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥))))))) = × )
5150opeq2d 4835 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩ = ⟹(.r‘ndx), × ⟩)
5227, 37, 51tpeq123d 4707 . . . . . . . 8 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} = {⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩})
5328opeq2d 4835 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟹(Scalar‘ndx), 𝑟⟩ = ⟹(Scalar‘ndx), 𝑅⟩)
5417adantr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Base‘𝑟) = 𝐟)
5543ofeqd 7611 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (.r‘𝑟) = ∘f · )
5638xpeq1d 5660 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {𝑥}) = (𝐷 × {𝑥}))
57 eqidd 2738 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
5855, 56, 57oveq123d 7372 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓) = ((𝐷 × {𝑥}) ∘f · 𝑓))
5954, 26, 58mpoeq123dv 7426 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓)) = (𝑥 ∈ 𝐟, 𝑓 ∈ 𝐵 ↩ ((𝐷 × {𝑥}) ∘f · 𝑓)))
60 psrval.v . . . . . . . . . . 11 ∙ = (𝑥 ∈ 𝐟, 𝑓 ∈ 𝐵 ↩ ((𝐷 × {𝑥}) ∘f · 𝑓))
6159, 60eqtr4di 2795 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓)) = ∙ )
6261opeq2d 4835 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩ = ⟹( ·𝑠 ‘ndx), ∙ ⟩)
6328fveq2d 6843 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = (TopOpen‘𝑅))
64 psrval.o . . . . . . . . . . . . . . 15 𝑂 = (TopOpen‘𝑅)
6563, 64eqtr4di 2795 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = 𝑂)
6665sneqd 4596 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {(TopOpen‘𝑟)} = {𝑂})
6738, 66xpeq12d 5662 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {(TopOpen‘𝑟)}) = (𝐷 × {𝑂}))
6867fveq2d 6843 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = (∏t‘(𝐷 × {𝑂})))
69 psrval.j . . . . . . . . . . . 12 (𝜑 → 𝐜 = (∏t‘(𝐷 × {𝑂})))
7069ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝐜 = (∏t‘(𝐷 × {𝑂})))
7168, 70eqtr4d 2780 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = 𝐜)
7271opeq2d 4835 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ = ⟹(TopSet‘ndx), 𝐜⟩)
7353, 62, 72tpeq123d 4707 . . . . . . . 8 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩} = {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩})
7452, 73uneq12d 4122 . . . . . . 7 ((((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
7525, 74csbied 3891 . . . . . 6 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ⊋𝐵 / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
7623, 75eqtrd 2777 . . . . 5 (((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ⩋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
7713, 76csbied 3891 . . . 4 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → ⊋𝐷 / 𝑑⊌⊋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
7810, 77eqtrd 2777 . . 3 ((𝜑 ∧ (𝑖 = 𝐌 ∧ 𝑟 = 𝑅)) → ⩋{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⊌⊋((Base‘𝑟) ↑m 𝑑) / 𝑏⊌({⟹(Base‘ndx), 𝑏⟩, ⟹(+g‘ndx), ( ∘f (+g‘𝑟) ↟ (𝑏 × 𝑏))⟩, ⟹(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↩ (𝑘 ∈ 𝑑 ↩ (𝑟 Σg (𝑥 ∈ {𝑊 ∈ 𝑑 ∣ 𝑊 ∘r ≀ 𝑘} ↩ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))⟩} ∪ {⟹(Scalar‘ndx), 𝑟⟩, ⟹( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↩ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))⟩, ⟹(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
79 psrval.i . . . 4 (𝜑 → 𝐌 ∈ 𝑊)
8079elexd 3463 . . 3 (𝜑 → 𝐌 ∈ V)
81 psrval.r . . . 4 (𝜑 → 𝑅 ∈ 𝑋)
8281elexd 3463 . . 3 (𝜑 → 𝑅 ∈ V)
83 tpex 7673 . . . . 5 {⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∈ V
84 tpex 7673 . . . . 5 {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩} ∈ V
8583, 84unex 7672 . . . 4 ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}) ∈ V
8685a1i 11 . . 3 (𝜑 → ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}) ∈ V)
873, 78, 80, 82, 86ovmpod 7501 . 2 (𝜑 → (𝐌 mPwSer 𝑅) = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
881, 87eqtrid 2789 1 (𝜑 → 𝑆 = ({⟹(Base‘ndx), 𝐵⟩, ⟹(+g‘ndx), ✚ ⟩, ⟹(.r‘ndx), × ⟩} ∪ {⟹(Scalar‘ndx), 𝑅⟩, ⟹( ·𝑠 ‘ndx), ∙ ⟩, ⟹(TopSet‘ndx), 𝐜⟩}))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3405  Vcvv 3443  âŠ‹csb 3853   ∪ cun 3906  {csn 4584  {ctp 4588  âŸšcop 4590   class class class wbr 5103   ↩ cmpt 5186   × cxp 5629  â—¡ccnv 5630   ↟ cres 5633   “ cima 5634  â€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353   ∘f cof 7607   ∘r cofr 7608   ↑m cmap 8723  Fincfn 8841   ≀ cle 11148   − cmin 11343  â„•cn 12111  â„•0cn0 12371  ndxcnx 17025  Basecbs 17043  +gcplusg 17093  .rcmulr 17094  Scalarcsca 17096   ·𝑠 cvsca 17097  TopSetcts 17099  TopOpenctopn 17263  âˆtcpt 17280   Σg cgsu 17282   mPwSer cmps 21259
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609  df-psr 21264
This theorem is referenced by:  psrbas  21299  psrplusg  21302  psrmulr  21305  psrsca  21310  psrvscafval  21311
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