Theorem selvffval 22340

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)

Assertion

Ref	Expression
selvffval	⊢ (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))))))))

Distinct variable groups:   𝑗,𝐼,𝑓,𝑢,𝑡,𝑐,𝑑,𝑥   𝑅,𝑗,𝑓,𝑢,𝑡,𝑐,𝑑,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑢, 𝑡, 𝑓, 𝑗, 𝑐, 𝑑)   𝑉(𝑥, 𝑢, 𝑡, 𝑓, 𝑗, 𝑐, 𝑑)   𝑊(𝑥, 𝑢, 𝑡, 𝑓, 𝑗, 𝑐, 𝑑)

Proof of Theorem selvffval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-selv 22339	. . 3 ⊢ selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (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))))))))
2	1	a1i 11	. 2 ⊢ (𝜑 → selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (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)))))))))
3		pweq 4574	. . . . 5 ⊢ (𝑖 = 𝐼 → 𝒫 𝑖 = 𝒫 𝐼)
4	3	adantr 486	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝒫 𝑖 = 𝒫 𝐼)
5		oveq12 7426	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
6	5	fveq2d 6886	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘(𝐼 mPoly 𝑅)))
7		difeq1 4070	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ 𝑗) = (𝐼 ∖ 𝑗))
8	7	adantr 486	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 ∖ 𝑗) = (𝐼 ∖ 𝑗))
9		simpr 490	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
10	8, 9	oveq12d 7435	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ 𝑗) mPoly 𝑟) = ((𝐼 ∖ 𝑗) mPoly 𝑅))
11		oveq1 7424	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (𝑖 evalSub 𝑡) = (𝐼 evalSub 𝑡))
12	11	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑡) = (𝐼 evalSub 𝑡))
13	12	fveq1d 6884	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑡)‘ran 𝑑) = ((𝐼 evalSub 𝑡)‘ran 𝑑))
14	13	fveq1d 6884	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (((𝑖 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓)) = (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓)))
15		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
16	8, 9	oveq12d 7435	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ 𝑗) mVar 𝑟) = ((𝐼 ∖ 𝑗) mVar 𝑅))
17	16	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥) = (((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))
18	17	fveq2d 6886	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑐‘(((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥)) = (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))
19	18	ifeq2d 4506	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 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))))
20	15, 19	mpteq12dv 5196	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (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)))))
21	14, 20	fveq12d 6889	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((((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))))))
22	21	csbeq2dv 3857	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(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))))))
23	22	csbeq2dv 3857	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(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))))))
24	23	csbeq2dv 3857	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ⦋(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))))))
25	10, 24	csbeq12dv 3859	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 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))))) = ⦋((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))))))
26	6, 25	mpteq12dv 5196	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (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)))))))
27	4, 26	mpteq12dv 5196	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 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))))))) = (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))))))))
28	27	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 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))))))) = (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))))))))
29		selvffval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
30	29	elexd 3476	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
31		selvffval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
32	31	elexd 3476	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
33	29	pwexd 5348	. . 3 ⊢ (𝜑 → 𝒫 𝐼 ∈ V)
34	33	mptexd 7227	. 2 ⊢ (𝜑 → (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))))))) ∈ V)
35	2, 28, 30, 32, 34	ovmpod 7569	1 ⊢ (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))))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⦋csb 3850   ∖ cdif 3899  ifcif 4485  𝒫 cpw 4560   ↦ cmpt 5190  ran crn 5660   ∘ ccom 5663  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Basecbs 17307  algSccascl 22073   mVar cmvr 22126   mPoly cmpl 22127   evalSub ces 22294   selectVars cslv 22338

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-selv 22339

This theorem is used by:  selvfval  22341