msrfval - Mathbox for Mario Carneiro

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Theorem msrfval 35564

Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
msrfval.v	⊢ 𝑉 = (mVars‘𝑇)
msrfval.p	⊢ 𝑃 = (mPreSt‘𝑇)
msrfval.r	⊢ 𝑅 = (mStRed‘𝑇)

**Assertion**
Ref	Expression
msrfval	⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)

Distinct variable groups: ℎ,𝑎,𝑠,𝑧,𝑃 𝑇,𝑎,ℎ,𝑠 𝑧,𝑉 Allowed substitution hints: 𝑅(𝑧,ℎ,𝑠,𝑎) 𝑇(𝑧) 𝑉(ℎ,𝑠,𝑎)

Proof of Theorem msrfval Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		msrfval.r	. 2 ⊢ 𝑅 = (mStRed‘𝑇)
2		fveq2 6881	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		msrfval.p	. . . . . 6 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = (mVars‘𝑇))
6		msrfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (mVars‘𝑇)
7	5, 6	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (mVars‘𝑡) = 𝑉)
8	7	imaeq1d 6051	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = (𝑉 “ (ℎ ∪ {𝑎})))
9	8	unieqd 4901	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) = ∪ (𝑉 “ (ℎ ∪ {𝑎})))
10	9	csbeq1d 3883	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧) = ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧))
11	10	ineq2d 4200	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)) = ((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)))
12	11	oteq1d 4866	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
13	12	csbeq2dv 3886	. . . . . 6 ⊢ (𝑡 = 𝑇 → ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
14	13	csbeq2dv 3886	. . . . 5 ⊢ (𝑡 = 𝑇 → ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩ = ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
15	4, 14	mpteq12dv 5212	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
16		df-msr 35521	. . . 4 ⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
17	15, 16, 3	mptfvmpt 7225	. . 3 ⊢ (𝑇 ∈ V → (mStRed‘𝑇) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
18		mpt0 6685	. . . . 5 ⊢ (𝑠 ∈ ∅ ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩) = ∅
19	18	eqcomi 2745	. . . 4 ⊢ ∅ = (𝑠 ∈ ∅ ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
20		fvprc 6873	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
21		fvprc 6873	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
22	3, 21	eqtrid 2783	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑃 = ∅)
23	22	mpteq1d 5215	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩) = (𝑠 ∈ ∅ ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
24	19, 20, 23	3eqtr4a 2797	. . 3 ⊢ (¬ 𝑇 ∈ V → (mStRed‘𝑇) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
25	17, 24	pm2.61i 182	. 2 ⊢ (mStRed‘𝑇) = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)
26	1, 25	eqtri 2759	1 ⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ⦋csb 3879 ∪ cun 3929 ∩ cin 3930 ∅c0 4313 {csn 4606 ⟨cotp 4614 ∪ cuni 4888 ↦ cmpt 5206 × cxp 5657 “ cima 5662 ‘cfv 6536 1^st c1st 7991 2^nd c2nd 7992 mVarscmvrs 35496 mPreStcmpst 35500 mStRedcmsr 35501

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pr 5407

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-ot 4615 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 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 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-msr 35521

This theorem is referenced by: msrval 35565 msrf 35569

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