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Theorem selvfval 21679
Description: Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
Hypotheses
Ref Expression
selvffval.i (πœ‘ β†’ 𝐼 ∈ 𝑉)
selvffval.r (πœ‘ β†’ 𝑅 ∈ π‘Š)
selvfval.j (πœ‘ β†’ 𝐽 βŠ† 𝐼)
Assertion
Ref Expression
selvfval (πœ‘ β†’ ((𝐼 selectVars 𝑅)β€˜π½) = (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))))
Distinct variable groups:   𝑓,𝐼,𝑒,𝑑,𝑐,𝑑,π‘₯   𝑅,𝑓,𝑒,𝑑,𝑐,𝑑,π‘₯   𝑓,𝐽,𝑒,𝑑,𝑐,𝑑,π‘₯
Allowed substitution hints:   πœ‘(π‘₯,𝑒,𝑑,𝑓,𝑐,𝑑)   𝑉(π‘₯,𝑒,𝑑,𝑓,𝑐,𝑑)   π‘Š(π‘₯,𝑒,𝑑,𝑓,𝑐,𝑑)

Proof of Theorem selvfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 selvffval.i . . 3 (πœ‘ β†’ 𝐼 ∈ 𝑉)
2 selvffval.r . . 3 (πœ‘ β†’ 𝑅 ∈ π‘Š)
31, 2selvffval 21678 . 2 (πœ‘ β†’ (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))))
4 difeq2 4116 . . . . . 6 (𝑗 = 𝐽 β†’ (𝐼 βˆ– 𝑗) = (𝐼 βˆ– 𝐽))
54oveq1d 7423 . . . . 5 (𝑗 = 𝐽 β†’ ((𝐼 βˆ– 𝑗) mPoly 𝑅) = ((𝐼 βˆ– 𝐽) mPoly 𝑅))
6 oveq1 7415 . . . . . 6 (𝑗 = 𝐽 β†’ (𝑗 mPoly 𝑒) = (𝐽 mPoly 𝑒))
7 eleq2 2822 . . . . . . . . . . 11 (𝑗 = 𝐽 β†’ (π‘₯ ∈ 𝑗 ↔ π‘₯ ∈ 𝐽))
8 oveq1 7415 . . . . . . . . . . . 12 (𝑗 = 𝐽 β†’ (𝑗 mVar 𝑒) = (𝐽 mVar 𝑒))
98fveq1d 6893 . . . . . . . . . . 11 (𝑗 = 𝐽 β†’ ((𝑗 mVar 𝑒)β€˜π‘₯) = ((𝐽 mVar 𝑒)β€˜π‘₯))
104oveq1d 7423 . . . . . . . . . . . . 13 (𝑗 = 𝐽 β†’ ((𝐼 βˆ– 𝑗) mVar 𝑅) = ((𝐼 βˆ– 𝐽) mVar 𝑅))
1110fveq1d 6893 . . . . . . . . . . . 12 (𝑗 = 𝐽 β†’ (((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯) = (((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))
1211fveq2d 6895 . . . . . . . . . . 11 (𝑗 = 𝐽 β†’ (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)) = (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))
137, 9, 12ifbieq12d 4556 . . . . . . . . . 10 (𝑗 = 𝐽 β†’ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))) = if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))
1413mpteq2dv 5250 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)))) = (π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))
1514fveq2d 6895 . . . . . . . 8 (𝑗 = 𝐽 β†’ ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))) = ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))))
1615csbeq2dv 3900 . . . . . . 7 (𝑗 = 𝐽 β†’ ⦋(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))) = ⦋(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))))
1716csbeq2dv 3900 . . . . . 6 (𝑗 = 𝐽 β†’ ⦋(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))) = ⦋(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))))
186, 17csbeq12dv 3902 . . . . 5 (𝑗 = 𝐽 β†’ ⦋(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))) = ⦋(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))))
195, 18csbeq12dv 3902 . . . 4 (𝑗 = 𝐽 β†’ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))) = ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))))
2019mpteq2dv 5250 . . 3 (𝑗 = 𝐽 β†’ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)))))) = (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))))
2120adantl 482 . 2 ((πœ‘ ∧ 𝑗 = 𝐽) β†’ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)))))) = (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))))
22 selvfval.j . . 3 (πœ‘ β†’ 𝐽 βŠ† 𝐼)
231, 22sselpwd 5326 . 2 (πœ‘ β†’ 𝐽 ∈ 𝒫 𝐼)
24 fvex 6904 . . 3 (Baseβ€˜(𝐼 mPoly 𝑅)) ∈ V
25 mptexg 7222 . . 3 ((Baseβ€˜(𝐼 mPoly 𝑅)) ∈ V β†’ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))) ∈ V)
2624, 25mp1i 13 . 2 (πœ‘ β†’ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))) ∈ V)
273, 21, 23, 26fvmptd 7005 1 (πœ‘ β†’ ((𝐼 selectVars 𝑅)β€˜π½) = (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝐽) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝐽 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯)))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  β¦‹csb 3893   βˆ– cdif 3945   βŠ† wss 3948  ifcif 4528  π’« cpw 4602   ↦ cmpt 5231  ran crn 5677   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  algSccascl 21406   mVar cmvr 21457   mPoly cmpl 21458   evalSub ces 21632   selectVars cslv 21670
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-selv 21674
This theorem is referenced by:  selvval  21680
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