Theorem selvffval 21236

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 21232	. . 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 4546	. . . . 5 ⊢ (𝑖 = 𝐼 → 𝒫 𝑖 = 𝒫 𝐼)
4	3	adantr 480	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝒫 𝑖 = 𝒫 𝐼)
5		oveq12 7264	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
6	5	fveq2d 6760	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘(𝐼 mPoly 𝑅)))
7		difeq1 4046	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑖 ∖ 𝑗) = (𝐼 ∖ 𝑗))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 ∖ 𝑗) = (𝐼 ∖ 𝑗))
9		simpr 484	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
10	8, 9	oveq12d 7273	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ 𝑗) mPoly 𝑟) = ((𝐼 ∖ 𝑗) mPoly 𝑅))
11		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (𝑖 evalSub 𝑡) = (𝐼 evalSub 𝑡))
12	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑡) = (𝐼 evalSub 𝑡))
13	12	fveq1d 6758	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑡)‘ran 𝑑) = ((𝐼 evalSub 𝑡)‘ran 𝑑))
14	13	fveq1d 6758	. . . . . . . . . 10 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (((𝑖 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓)) = (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓)))
15		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
16	8, 9	oveq12d 7273	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 ∖ 𝑗) mVar 𝑟) = ((𝐼 ∖ 𝑗) mVar 𝑅))
17	16	fveq1d 6758	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥) = (((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))
18	17	fveq2d 6760	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑐‘(((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥)) = (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))
19	18	ifeq2d 4476	. . . . . . . . . . 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 5161	. . . . . . . . . 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 6763	. . . . . . . . 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 3835	. . . . . . . 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 3835	. . . . . . 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 3835	. . . . . 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 3837	. . . . 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 5161	. . . 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 5161	. . 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 481	. 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 3442	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
31		selvffval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
32	31	elexd 3442	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
33	29	pwexd 5297	. . 3 ⊢ (𝜑 → 𝒫 𝐼 ∈ V)
34	33	mptexd 7082	. 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 7403	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))))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⦋csb 3828   ∖ cdif 3880  ifcif 4456  𝒫 cpw 4530   ↦ cmpt 5153  ran crn 5581   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  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:  selvfval  21237