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Theorem selvfval 22077

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 22076	. 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 4116	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝐼 ∖ 𝑗) = (𝐼 ∖ 𝐽))
5	4	oveq1d 7441	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝐼 ∖ 𝑗) mPoly 𝑅) = ((𝐼 ∖ 𝐽) mPoly 𝑅))
6		oveq1 7433	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑗 mPoly 𝑢) = (𝐽 mPoly 𝑢))
7		eleq2 2818	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ 𝐽))
8		oveq1 7433	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝑗 mVar 𝑢) = (𝐽 mVar 𝑢))
9	8	fveq1d 6904	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝑗 mVar 𝑢)‘𝑥) = ((𝐽 mVar 𝑢)‘𝑥))
10	4	oveq1d 7441	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → ((𝐼 ∖ 𝑗) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅))
11	10	fveq1d 6904	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥) = (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))
12	11	fveq2d 6906	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)) = (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))
13	7, 9, 12	ifbieq12d 4560	. . . . . . . . . 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 5254	. . . . . . . . 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 6906	. . . . . . . 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 3901	. . . . . . 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 3901	. . . . . 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 3903	. . . . 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 3903	. . . 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 5254	. . 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 480	. 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 5332	. 2 ⊢ (𝜑 → 𝐽 ∈ 𝒫 𝐼)
24		fvex 6915	. . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) ∈ V
25		mptexg 7239	. . 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 7017	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)))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⦋csb 3894 ∖ cdif 3946 ⊆ wss 3949 ifcif 4532 𝒫 cpw 4606 ↦ cmpt 5235 ran crn 5683 ∘ ccom 5686 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 algSccascl 21800 mVar cmvr 21852 mPoly cmpl 21853 evalSub ces 22033 selectVars cslv 22071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-selv 22075

This theorem is referenced by: selvval 22078

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