Theorem selvfval 21237

Description: Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)

Hypotheses

Ref	Expression
selvffval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
selvffval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
selvfval.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)

Assertion

Ref	Expression
selvfval	⊢ (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) 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.

Step	Hyp	Ref	Expression
1		selvffval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		selvffval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
3	1, 2	selvffval 21236	. 2 ⊢ (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))))))
4		difeq2 4047	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝐼 ∖ 𝑗) = (𝐼 ∖ 𝐽))
5	4	oveq1d 7270	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝐼 ∖ 𝑗) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅))
6		oveq1 7262	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑗 mPoly 𝑢) = (𝐽 mPoly 𝑢))
7		eleq2 2827	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ 𝐽))
8		oveq1 7262	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝑗 mVar 𝑢) = (𝐽 mVar 𝑢))
9	8	fveq1d 6758	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝑗 mVar 𝑢)‘𝑥) = ((𝐽 mVar 𝑢)‘𝑥))
10	4	oveq1d 7270	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → ((𝐼 ∖ 𝑗) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅))
11	10	fveq1d 6758	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥) = (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))
12	11	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)) = (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))
13	7, 9, 12	ifbieq12d 4484	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))) = if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))
14	13	mpteq2dv 5172	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))
15	14	fveq2d 6760	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))) = ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
16	15	csbeq2dv 3835	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))) = ⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
17	16	csbeq2dv 3835	. . . . . 6 ⊢ (𝑗 = 𝐽 → ⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))) = ⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
18	6, 17	csbeq12dv 3837	. . . . 5 ⊢ (𝑗 = 𝐽 → ⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))) = ⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
19	5, 18	csbeq12dv 3837	. . . 4 ⊢ (𝑗 = 𝐽 → ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))) = ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
20	19	mpteq2dv 5172	. . 3 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))))) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))))
21	20	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))))) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))))
22		selvfval.j	. . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
23	1, 22	sselpwd 5245	. 2 ⊢ (𝜑 → 𝐽 ∈ 𝒫 𝐼)
24		fvex 6769	. . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) ∈ V
25		mptexg 7079	. . 3 ⊢ ((Base‘(𝐼 mPoly 𝑅)) ∈ V → (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))) ∈ V)
26	24, 25	mp1i 13	. 2 ⊢ (𝜑 → (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))) ∈ V)
27	3, 21, 23, 26	fvmptd 6864	1 ⊢ (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⦋csb 3828   ∖ cdif 3880   ⊆ wss 3883  ifcif 4456  𝒫 cpw 4530   ↦ cmpt 5153  ran crn 5581   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  algSccascl 20969   mVar cmvr 21018   mPoly cmpl 21019   evalSub ces 21190   selectVars cslv 21228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-selv 21232

This theorem is referenced by:  selvval  21238