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Theorem selvval 21528
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 21527 . . 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 5814 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑑𝑓) = (𝑑𝐹))
65fveq2d 6846 . . . . . . . . 9 (𝑓 = 𝐹 → (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓)) = (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹)))
76fveq1d 6844 . . . . . . . 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 3862 . . . . . . 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 3862 . . . . . 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 3862 . . . . 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 3862 . . . 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 6845 . . . . 5 (Base‘𝑃) = (Base‘(𝐼 mPoly 𝑅))
1714, 16eqtri 2764 . . . 4 𝐵 = (Base‘(𝐼 mPoly 𝑅))
1813, 17eleqtrdi 2848 . . 3 (𝜑𝐹 ∈ (Base‘(𝐼 mPoly 𝑅)))
19 fvex 6855 . . . . . . . 8 ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))) ∈ V
2019csbex 5268 . . . . . . 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 5268 . . . . . 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 5268 . . . . 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 5268 . . . 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 6955 . 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 7390 . . 3 ((𝐼𝐽) mPoly 𝑅) ∈ V
27 selvval.u . . . . 5 𝑈 = ((𝐼𝐽) mPoly 𝑅)
2827eqeq2i 2749 . . . 4 (𝑢 = 𝑈𝑢 = ((𝐼𝐽) mPoly 𝑅))
29 oveq2 7365 . . . . . 6 (𝑢 = 𝑈 → (𝐽 mPoly 𝑢) = (𝐽 mPoly 𝑈))
30 fveq2 6842 . . . . . . . . 9 (𝑢 = 𝑈 → (algSc‘𝑢) = (algSc‘𝑈))
3130coeq2d 5818 . . . . . . . 8 (𝑢 = 𝑈 → (𝑐 ∘ (algSc‘𝑢)) = (𝑐 ∘ (algSc‘𝑈)))
32 oveq2 7365 . . . . . . . . . . . 12 (𝑢 = 𝑈 → (𝐽 mVar 𝑢) = (𝐽 mVar 𝑈))
3332fveq1d 6844 . . . . . . . . . . 11 (𝑢 = 𝑈 → ((𝐽 mVar 𝑢)‘𝑥) = ((𝐽 mVar 𝑈)‘𝑥))
3433ifeq1d 4505 . . . . . . . . . 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 5207 . . . . . . . . 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 6846 . . . . . . . 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 3864 . . . . . . 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 3862 . . . . . 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 3864 . . . . 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 7390 . . . . . 6 (𝐽 mPoly 𝑈) ∈ V
41 selvval.t . . . . . . . 8 𝑇 = (𝐽 mPoly 𝑈)
4241eqeq2i 2749 . . . . . . 7 (𝑡 = 𝑇𝑡 = (𝐽 mPoly 𝑈))
43 fveq2 6842 . . . . . . . . 9 (𝑡 = 𝑇 → (algSc‘𝑡) = (algSc‘𝑇))
44 oveq2 7365 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (𝐼 evalSub 𝑡) = (𝐼 evalSub 𝑇))
4544fveq1d 6844 . . . . . . . . . . . 12 (𝑡 = 𝑇 → ((𝐼 evalSub 𝑡)‘ran 𝑑) = ((𝐼 evalSub 𝑇)‘ran 𝑑))
4645fveq1d 6844 . . . . . . . . . . 11 (𝑡 = 𝑇 → (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝐹)) = (((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹)))
4746fveq1d 6844 . . . . . . . . . 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 3862 . . . . . . . . 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 3864 . . . . . . . 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 6855 . . . . . . . . 9 (algSc‘𝑇) ∈ V
51 selvval.c . . . . . . . . . . 11 𝐶 = (algSc‘𝑇)
5251eqeq2i 2749 . . . . . . . . . 10 (𝑐 = 𝐶𝑐 = (algSc‘𝑇))
53 coeq1 5813 . . . . . . . . . . . 12 (𝑐 = 𝐶 → (𝑐 ∘ (algSc‘𝑈)) = (𝐶 ∘ (algSc‘𝑈)))
54 fveq1 6841 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)) = (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))
5554ifeq2d 4506 . . . . . . . . . . . . . 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 5207 . . . . . . . . . . . . 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 6846 . . . . . . . . . . . 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 3864 . . . . . . . . . . 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 6856 . . . . . . . . . . . . 13 𝐶 ∈ V
60 fvex 6855 . . . . . . . . . . . . 13 (algSc‘𝑈) ∈ V
6159, 60coex 7867 . . . . . . . . . . . 12 (𝐶 ∘ (algSc‘𝑈)) ∈ V
62 selvval.d . . . . . . . . . . . . . 14 𝐷 = (𝐶 ∘ (algSc‘𝑈))
6362eqeq2i 2749 . . . . . . . . . . . . 13 (𝑑 = 𝐷𝑑 = (𝐶 ∘ (algSc‘𝑈)))
64 rneq 5891 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → ran 𝑑 = ran 𝐷)
6564fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → ((𝐼 evalSub 𝑇)‘ran 𝑑) = ((𝐼 evalSub 𝑇)‘ran 𝐷))
66 coeq1 5813 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → (𝑑𝐹) = (𝐷𝐹))
6765, 66fveq12d 6849 . . . . . . . . . . . . . 14 (𝑑 = 𝐷 → (((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑𝐹)) = (((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹)))
6867fveq1d 6844 . . . . . . . . . . . . 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 3891 . . . . . . . . . . 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 3891 . . . . . . . 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 3891 . . . . 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 3891 . 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 1541  wcel 2106  Vcvv 3445  csb 3855  cdif 3907  wss 3910  ifcif 4486  cmpt 5188  ran crn 5634  ccom 5637  cfv 6496  (class class class)co 7357  Basecbs 17083  algSccascl 21258   mVar cmvr 21307   mPoly cmpl 21308   evalSub ces 21480   selectVars cslv 21518
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-selv 21522
This theorem is referenced by:  selvcl  40748  selvval2  40749
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