Theorem selvval 22103

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

Hypotheses

Ref	Expression
selvval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
selvval.b	⊢ 𝐵 = (Base‘𝑃)
selvval.u	⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
selvval.t	⊢ 𝑇 = (𝐽 mPoly 𝑈)
selvval.c	⊢ 𝐶 = (algSc‘𝑇)
selvval.d	⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))
selvval.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)
selvval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

Assertion

Ref	Expression
selvval	⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((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:   𝑥,𝐼   𝑥,𝑅   𝑥,𝐽   𝑥,𝑈   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐷(𝑥)   𝑃(𝑥)   𝑇(𝑥)   𝐹(𝑥)

Proof of Theorem selvval

Dummy variables 𝑓 𝑢 𝑡 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coeq2 5807	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑑 ∘ 𝑓) = (𝑑 ∘ 𝐹))
2	1	fveq2d 6838	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓)) = (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝐹)))
3	2	fveq1d 6836	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((((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))))))
4	3	csbeq2dv 3845	. . . . . 6 ⊢ (𝑓 = 𝐹 → ⦋(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))))))
5	4	csbeq2dv 3845	. . . . 5 ⊢ (𝑓 = 𝐹 → ⦋(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))))))
6	5	csbeq2dv 3845	. . . 4 ⊢ (𝑓 = 𝐹 → ⦋(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))))))
7	6	csbeq2dv 3845	. . 3 ⊢ (𝑓 = 𝐹 → ⦋((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))))))
8		selvval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
9		reldmmpl 21969	. . . . . . 7 ⊢ Rel dom mPoly
10		selvval.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
11		selvval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
12	9, 10, 11	elbasov 17184	. . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
14	13	simpld 495	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
15	13	simprd 496	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
16		selvval.j	. . . 4 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
17	14, 15, 16	selvfval 22102	. . 3 ⊢ (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)))))))
18	10	fveq2i 6837	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(𝐼 mPoly 𝑅))
19	11, 18	eqtri 2763	. . . 4 ⊢ 𝐵 = (Base‘(𝐼 mPoly 𝑅))
20	8, 19	eleqtrdi 2850	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly 𝑅)))
21		fvex 6847	. . . . . . . 8 ⊢ ((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))) ∈ V
22	21	csbex 5240	. . . . . . 7 ⊢ ⦋(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
23	22	csbex 5240	. . . . . 6 ⊢ ⦋(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
24	23	csbex 5240	. . . . 5 ⊢ ⦋(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
25	24	csbex 5240	. . . 4 ⊢ ⦋((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	25	a1i 11	. . 3 ⊢ (𝜑 → ⦋((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	7, 17, 20, 26	fvmptd4 6967	. 2 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ⦋((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		ovex 7396	. . 3 ⊢ ((𝐼 ∖ 𝐽) mPoly 𝑅) ∈ V
29		selvval.u	. . . . 5 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
30	29	eqeq2i 2753	. . . 4 ⊢ (𝑢 = 𝑈 ↔ 𝑢 = ((𝐼 ∖ 𝐽) mPoly 𝑅))
31		oveq2 7371	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝐽 mPoly 𝑢) = (𝐽 mPoly 𝑈))
32		fveq2 6834	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (algSc‘𝑢) = (algSc‘𝑈))
33	32	coeq2d 5811	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑐 ∘ (algSc‘𝑢)) = (𝑐 ∘ (algSc‘𝑈)))
34		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑈 → (𝐽 mVar 𝑢) = (𝐽 mVar 𝑈))
35	34	fveq1d 6836	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ((𝐽 mVar 𝑢)‘𝑥) = ((𝐽 mVar 𝑈)‘𝑥))
36	35	ifeq1d 4481	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → 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))))
37	36	mpteq2dv 5173	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (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)))))
38	37	fveq2d 6838	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((((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))))))
39	33, 38	csbeq12dv 3847	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ⦋(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))))))
40	39	csbeq2dv 3845	. . . . . 6 ⊢ (𝑢 = 𝑈 → ⦋(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))))))
41	31, 40	csbeq12dv 3847	. . . . 5 ⊢ (𝑢 = 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 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))))))
42		ovex 7396	. . . . . 6 ⊢ (𝐽 mPoly 𝑈) ∈ V
43		selvval.t	. . . . . . . 8 ⊢ 𝑇 = (𝐽 mPoly 𝑈)
44	43	eqeq2i 2753	. . . . . . 7 ⊢ (𝑡 = 𝑇 ↔ 𝑡 = (𝐽 mPoly 𝑈))
45		fveq2 6834	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (algSc‘𝑡) = (algSc‘𝑇))
46		oveq2 7371	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (𝐼 evalSub 𝑡) = (𝐼 evalSub 𝑇))
47	46	fveq1d 6836	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → ((𝐼 evalSub 𝑡)‘ran 𝑑) = ((𝐼 evalSub 𝑇)‘ran 𝑑))
48	47	fveq1d 6836	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝐹)) = (((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑 ∘ 𝐹)))
49	48	fveq1d 6836	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((((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))))))
50	49	csbeq2dv 3845	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋(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))))))
51	45, 50	csbeq12dv 3847	. . . . . . . 8 ⊢ (𝑡 = 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))))) = ⦋(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))))))
52		fvex 6847	. . . . . . . . 9 ⊢ (algSc‘𝑇) ∈ V
53		selvval.c	. . . . . . . . . . 11 ⊢ 𝐶 = (algSc‘𝑇)
54	53	eqeq2i 2753	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 ↔ 𝑐 = (algSc‘𝑇))
55		coeq1 5806	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → (𝑐 ∘ (algSc‘𝑈)) = (𝐶 ∘ (algSc‘𝑈)))
56		fveq1 6833	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝐶 → (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)) = (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))
57	56	ifeq2d 4482	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝐶 → 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))))
58	57	mpteq2dv 5173	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → (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)))))
59	58	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ((((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))))))
60	55, 59	csbeq12dv 3847	. . . . . . . . . . 11 ⊢ (𝑐 = 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))))) = ⦋(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))))))
61	53	fvexi 6848	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ V
62		fvex 6847	. . . . . . . . . . . . 13 ⊢ (algSc‘𝑈) ∈ V
63	61, 62	coex 7877	. . . . . . . . . . . 12 ⊢ (𝐶 ∘ (algSc‘𝑈)) ∈ V
64		selvval.d	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))
65	64	eqeq2i 2753	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝐷 ↔ 𝑑 = (𝐶 ∘ (algSc‘𝑈)))
66		rneq 5885	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝐷 → ran 𝑑 = ran 𝐷)
67	66	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝐷 → ((𝐼 evalSub 𝑇)‘ran 𝑑) = ((𝐼 evalSub 𝑇)‘ran 𝐷))
68		coeq1 5806	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝐷 → (𝑑 ∘ 𝐹) = (𝐷 ∘ 𝐹))
69	67, 68	fveq12d 6841	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝐷 → (((𝐼 evalSub 𝑇)‘ran 𝑑)‘(𝑑 ∘ 𝐹)) = (((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹)))
70	69	fveq1d 6836	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝐷 → ((((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))))))
71	65, 70	sylbir 236	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝐶 ∘ (algSc‘𝑈)) → ((((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))))))
72	63, 71	csbie 3873	. . . . . . . . . . 11 ⊢ ⦋(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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))
73	60, 72	eqtrdi 2791	. . . . . . . . . 10 ⊢ (𝑐 = 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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
74	54, 73	sylbir 236	. . . . . . . . 9 ⊢ (𝑐 = (algSc‘𝑇) → ⦋(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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
75	52, 74	csbie 3873	. . . . . . . 8 ⊢ ⦋(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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))
76	51, 75	eqtrdi 2791	. . . . . . 7 ⊢ (𝑡 = 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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
77	44, 76	sylbir 236	. . . . . 6 ⊢ (𝑡 = (𝐽 mPoly 𝑈) → ⦋(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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
78	42, 77	csbie 3873	. . . . 5 ⊢ ⦋(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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))
79	41, 78	eqtrdi 2791	. . . 4 ⊢ (𝑢 = 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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
80	30, 79	sylbir 236	. . 3 ⊢ (𝑢 = ((𝐼 ∖ 𝐽) mPoly 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))))) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
81	28, 80	csbie 3873	. 2 ⊢ ⦋((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 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))
82	27, 81	eqtrdi 2791	1 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((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 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ⦋csb 3838   ∖ cdif 3887   ⊆ wss 3890  ifcif 4461   ↦ cmpt 5160  ran crn 5626   ∘ ccom 5629  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  algSccascl 21834   mVar cmvr 21887   mPoly cmpl 21888   evalSub ces 22055   selectVars cslv 22099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-1cn 11094  ax-addcl 11096

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-nn 12173  df-slot 17150  df-ndx 17162  df-base 17178  df-mpl 21893  df-selv 22100

This theorem is referenced by:  selvcl  22123  selvval2  22124