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Theorem selvfval 21662
Description: Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
Hypotheses
Ref Expression
selvffval.i (𝜑𝐼𝑉)
selvffval.r (𝜑𝑅𝑊)
selvfval.j (𝜑𝐽𝐼)
Assertion
Ref Expression
selvfval (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)))))))
Distinct variable groups:   𝑓,𝐼,𝑢,𝑡,𝑐,𝑑,𝑥   𝑅,𝑓,𝑢,𝑡,𝑐,𝑑,𝑥   𝑓,𝐽,𝑢,𝑡,𝑐,𝑑,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑢,𝑡,𝑓,𝑐,𝑑)   𝑉(𝑥,𝑢,𝑡,𝑓,𝑐,𝑑)   𝑊(𝑥,𝑢,𝑡,𝑓,𝑐,𝑑)

Proof of Theorem selvfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 selvffval.i . . 3 (𝜑𝐼𝑉)
2 selvffval.r . . 3 (𝜑𝑅𝑊)
31, 2selvffval 21661 . 2 (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (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))))))))
4 difeq2 4115 . . . . . 6 (𝑗 = 𝐽 → (𝐼𝑗) = (𝐼𝐽))
54oveq1d 7419 . . . . 5 (𝑗 = 𝐽 → ((𝐼𝑗) mPoly 𝑅) = ((𝐼𝐽) mPoly 𝑅))
6 oveq1 7411 . . . . . 6 (𝑗 = 𝐽 → (𝑗 mPoly 𝑢) = (𝐽 mPoly 𝑢))
7 eleq2 2823 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝑥𝑗𝑥𝐽))
8 oveq1 7411 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (𝑗 mVar 𝑢) = (𝐽 mVar 𝑢))
98fveq1d 6890 . . . . . . . . . . 11 (𝑗 = 𝐽 → ((𝑗 mVar 𝑢)‘𝑥) = ((𝐽 mVar 𝑢)‘𝑥))
104oveq1d 7419 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → ((𝐼𝑗) mVar 𝑅) = ((𝐼𝐽) mVar 𝑅))
1110fveq1d 6890 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (((𝐼𝑗) mVar 𝑅)‘𝑥) = (((𝐼𝐽) mVar 𝑅)‘𝑥))
1211fveq2d 6892 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝑐‘(((𝐼𝑗) mVar 𝑅)‘𝑥)) = (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))
137, 9, 12ifbieq12d 4555 . . . . . . . . . 10 (𝑗 = 𝐽 → if(𝑥𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝑗) mVar 𝑅)‘𝑥))) = if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))
1413mpteq2dv 5249 . . . . . . . . 9 (𝑗 = 𝐽 → (𝑥𝐼 ↦ if(𝑥𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝑗) mVar 𝑅)‘𝑥)))) = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))
1514fveq2d 6892 . . . . . . . 8 (𝑗 = 𝐽 → ((((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))))))
1615csbeq2dv 3899 . . . . . . 7 (𝑗 = 𝐽(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))))))
1716csbeq2dv 3899 . . . . . 6 (𝑗 = 𝐽(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))))))
186, 17csbeq12dv 3901 . . . . 5 (𝑗 = 𝐽(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))))))
195, 18csbeq12dv 3901 . . . 4 (𝑗 = 𝐽((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))))))
2019mpteq2dv 5249 . . 3 (𝑗 = 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)))))) = (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)))))))
2120adantl 483 . 2 ((𝜑𝑗 = 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)))))) = (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)))))))
22 selvfval.j . . 3 (𝜑𝐽𝐼)
231, 22sselpwd 5325 . 2 (𝜑𝐽 ∈ 𝒫 𝐼)
24 fvex 6901 . . 3 (Base‘(𝐼 mPoly 𝑅)) ∈ V
25 mptexg 7218 . . 3 ((Base‘(𝐼 mPoly 𝑅)) ∈ V → (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)))))) ∈ V)
2624, 25mp1i 13 . 2 (𝜑 → (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)))))) ∈ V)
273, 21, 23, 26fvmptd 7001 1 (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)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  csb 3892  cdif 3944  wss 3947  ifcif 4527  𝒫 cpw 4601  cmpt 5230  ran crn 5676  ccom 5679  cfv 6540  (class class class)co 7404  Basecbs 17140  algSccascl 21391   mVar cmvr 21440   mPoly cmpl 21441   evalSub ces 21615   selectVars cslv 21653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-selv 21657
This theorem is referenced by:  selvval  21663
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