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Theorem selvval 21426
Description: Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
Hypotheses
Ref Expression
selvval.p 𝑃 = (𝐼 mPoly 𝑅)
selvval.b 𝐵 = (Base‘𝑃)
selvval.u 𝑈 = ((𝐼𝐽) mPoly 𝑅)
selvval.t 𝑇 = (𝐽 mPoly 𝑈)
selvval.c 𝐶 = (algSc‘𝑇)
selvval.d 𝐷 = (𝐶 ∘ (algSc‘𝑈))
selvval.i (𝜑𝐼𝑉)
selvval.r (𝜑𝑅𝑊)
selvval.j (𝜑𝐽𝐼)
selvval.f (𝜑𝐹𝐵)
Assertion
Ref Expression
selvval (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
Distinct variable groups:   𝑥,𝐼   𝑥,𝑅   𝑥,𝐽   𝑥,𝑈   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐷(𝑥)   𝑃(𝑥)   𝑇(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem selvval
Dummy variables 𝑓 𝑢 𝑡 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 selvval.i . . . 4 (𝜑𝐼𝑉)
2 selvval.r . . . 4 (𝜑𝑅𝑊)
3 selvval.j . . . 4 (𝜑𝐽𝐼)
41, 2, 3selvfval 21425 . . 3 (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))))
5 coeq2 5794 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑑𝑓) = (𝑑𝐹))
65fveq2d 6823 . . . . . . . . 9 (𝑓 = 𝐹 → (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓)) = (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹)))
76fveq1d 6821 . . . . . . . 8 (𝑓 = 𝐹 → ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
87csbeq2dv 3849 . . . . . . 7 (𝑓 = 𝐹(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
98csbeq2dv 3849 . . . . . 6 (𝑓 = 𝐹(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
109csbeq2dv 3849 . . . . 5 (𝑓 = 𝐹(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
1110csbeq2dv 3849 . . . 4 (𝑓 = 𝐹((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
1211adantl 482 . . 3 ((𝜑𝑓 = 𝐹) → ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
13 selvval.f . . . 4 (𝜑𝐹𝐵)
14 selvval.b . . . . 5 𝐵 = (Base‘𝑃)
15 selvval.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
1615fveq2i 6822 . . . . 5 (Base‘𝑃) = (Base‘(𝐼 mPoly 𝑅))
1714, 16eqtri 2764 . . . 4 𝐵 = (Base‘(𝐼 mPoly 𝑅))
1813, 17eleqtrdi 2847 . . 3 (𝜑𝐹 ∈ (Base‘(𝐼 mPoly 𝑅)))
19 fvex 6832 . . . . . . . 8 ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V
2019csbex 5252 . . . . . . 7 (𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V
2120csbex 5252 . . . . . 6 (algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V
2221csbex 5252 . . . . 5 (𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V
2322csbex 5252 . . . 4 ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V
2423a1i 11 . . 3 (𝜑((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V)
254, 12, 18, 24fvmptd 6932 . 2 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
26 ovex 7362 . . 3 ((𝐼𝐽) mPoly 𝑅) ∈ V
27 selvval.u . . . . 5 𝑈 = ((𝐼𝐽) mPoly 𝑅)
2827eqeq2i 2749 . . . 4 (𝑢 = 𝑈𝑢 = ((𝐼𝐽) mPoly 𝑅))
29 oveq2 7337 . . . . . 6 (𝑢 = 𝑈 → (𝐽 mPoly 𝑢) = (𝐽 mPoly 𝑈))
30 fveq2 6819 . . . . . . . . 9 (𝑢 = 𝑈 → (algSc‘𝑢) = (algSc‘𝑈))
3130coeq2d 5798 . . . . . . . 8 (𝑢 = 𝑈 → (𝑐 ∘ (algSc‘𝑢)) = (𝑐 ∘ (algSc‘𝑈)))
32 oveq2 7337 . . . . . . . . . . . 12 (𝑢 = 𝑈 → (𝐽 mVar 𝑢) = (𝐽 mVar 𝑈))
3332fveq1d 6821 . . . . . . . . . . 11 (𝑢 = 𝑈 → ((𝐽 mVar 𝑢)‘𝑥) = ((𝐽 mVar 𝑈)‘𝑥))
3433ifeq1d 4491 . . . . . . . . . 10 (𝑢 = 𝑈 → if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))) = if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))
3534mpteq2dv 5191 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))) = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
3635fveq2d 6823 . . . . . . . 8 (𝑢 = 𝑈 → ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
3731, 36csbeq12dv 3851 . . . . . . 7 (𝑢 = 𝑈(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
3837csbeq2dv 3849 . . . . . 6 (𝑢 = 𝑈(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
3929, 38csbeq12dv 3851 . . . . 5 (𝑢 = 𝑈(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (𝐽 mPoly 𝑈) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
40 ovex 7362 . . . . . 6 (𝐽 mPoly 𝑈) ∈ V
41 selvval.t . . . . . . . 8 𝑇 = (𝐽 mPoly 𝑈)
4241eqeq2i 2749 . . . . . . 7 (𝑡 = 𝑇𝑡 = (𝐽 mPoly 𝑈))
43 fveq2 6819 . . . . . . . . 9 (𝑡 = 𝑇 → (algSc‘𝑡) = (algSc‘𝑇))
44 oveq2 7337 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (𝐼 evalSub 𝑡) = (𝐼 evalSub 𝑇))
4544fveq1d 6821 . . . . . . . . . . . 12 (𝑡 = 𝑇 → ((𝐼 evalSub 𝑡)‘ran 𝑑) = ((𝐼 evalSub 𝑇)‘ran 𝑑))
4645fveq1d 6821 . . . . . . . . . . 11 (𝑡 = 𝑇 → (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹)) = (((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹)))
4746fveq1d 6821 . . . . . . . . . 10 (𝑡 = 𝑇 → ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
4847csbeq2dv 3849 . . . . . . . . 9 (𝑡 = 𝑇(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
4943, 48csbeq12dv 3851 . . . . . . . 8 (𝑡 = 𝑇(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (algSc‘𝑇) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
50 fvex 6832 . . . . . . . . 9 (algSc‘𝑇) ∈ V
51 selvval.c . . . . . . . . . . 11 𝐶 = (algSc‘𝑇)
5251eqeq2i 2749 . . . . . . . . . 10 (𝑐 = 𝐶𝑐 = (algSc‘𝑇))
53 coeq1 5793 . . . . . . . . . . . 12 (𝑐 = 𝐶 → (𝑐 ∘ (algSc‘𝑈)) = (𝐶 ∘ (algSc‘𝑈)))
54 fveq1 6818 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)) = (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))
5554ifeq2d 4492 . . . . . . . . . . . . . 14 (𝑐 = 𝐶 → if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))) = if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))
5655mpteq2dv 5191 . . . . . . . . . . . . 13 (𝑐 = 𝐶 → (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))) = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
5756fveq2d 6823 . . . . . . . . . . . 12 (𝑐 = 𝐶 → ((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
5853, 57csbeq12dv 3851 . . . . . . . . . . 11 (𝑐 = 𝐶(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = (𝐶 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
5951fvexi 6833 . . . . . . . . . . . . 13 𝐶 ∈ V
60 fvex 6832 . . . . . . . . . . . . 13 (algSc‘𝑈) ∈ V
6159, 60coex 7837 . . . . . . . . . . . 12 (𝐶 ∘ (algSc‘𝑈)) ∈ V
62 selvval.d . . . . . . . . . . . . . 14 𝐷 = (𝐶 ∘ (algSc‘𝑈))
6362eqeq2i 2749 . . . . . . . . . . . . 13 (𝑑 = 𝐷𝑑 = (𝐶 ∘ (algSc‘𝑈)))
64 rneq 5871 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → ran 𝑑 = ran 𝐷)
6564fveq2d 6823 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → ((𝐼 evalSub 𝑇)‘ran 𝑑) = ((𝐼 evalSub 𝑇)‘ran 𝐷))
66 coeq1 5793 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → (𝑑𝐹) = (𝐷𝐹))
6765, 66fveq12d 6826 . . . . . . . . . . . . . 14 (𝑑 = 𝐷 → (((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹)) = (((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹)))
6867fveq1d 6821 . . . . . . . . . . . . 13 (𝑑 = 𝐷 → ((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
6963, 68sylbir 234 . . . . . . . . . . . 12 (𝑑 = (𝐶 ∘ (algSc‘𝑈)) → ((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7061, 69csbie 3878 . . . . . . . . . . 11 (𝐶 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
7158, 70eqtrdi 2792 . . . . . . . . . 10 (𝑐 = 𝐶(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7252, 71sylbir 234 . . . . . . . . 9 (𝑐 = (algSc‘𝑇) → (𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7350, 72csbie 3878 . . . . . . . 8 (algSc‘𝑇) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
7449, 73eqtrdi 2792 . . . . . . 7 (𝑡 = 𝑇(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7542, 74sylbir 234 . . . . . 6 (𝑡 = (𝐽 mPoly 𝑈) → (algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7640, 75csbie 3878 . . . . 5 (𝐽 mPoly 𝑈) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑈)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
7739, 76eqtrdi 2792 . . . 4 (𝑢 = 𝑈(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7828, 77sylbir 234 . . 3 (𝑢 = ((𝐼𝐽) mPoly 𝑅) → (𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
7926, 78csbie 3878 . 2 ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
8025, 79eqtrdi 2792 1 (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3441  csb 3842  cdif 3894  wss 3897  ifcif 4472  cmpt 5172  ran crn 5615  ccom 5618  cfv 6473  (class class class)co 7329  Basecbs 17001  algSccascl 21157   mVar cmvr 21206   mPoly cmpl 21207   evalSub ces 21378   selectVars cslv 21416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-selv 21420
This theorem is referenced by:  selvcl  40477
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