resvval2 - Mathbox for Thierry Arnoux

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Theorem resvval2 29829

Description: Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)

**Hypotheses**
Ref	Expression
resvsca.r	⊢ 𝑅 = (𝑊 ↾_v 𝐴)
resvsca.f	⊢ 𝐹 = (Scalar‘𝑊)
resvsca.b	⊢ 𝐵 = (Base‘𝐹)

**Assertion**
Ref	Expression
resvval2	⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾_s 𝐴)⟩))

**Proof of Theorem resvval2**
Step	Hyp	Ref	Expression
1		resvsca.r	. . . 4 ⊢ 𝑅 = (𝑊 ↾_v 𝐴)
2		resvsca.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3		resvsca.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
4	1, 2, 3	resvval 29827	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾_s 𝐴)⟩)))
5		iffalse 4095	. . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾_s 𝐴)⟩)) = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾_s 𝐴)⟩))
6	4, 5	sylan9eqr 2678	. 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾_s 𝐴)⟩))
7	6	3impb 1260	1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾_s 𝐴)⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ifcif 4086 ⟨cop 4183 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 Basecbs 15857 ↾_s cress 15858 Scalarcsca 15944 ↾_v cresv 29824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-resv 29825

This theorem is referenced by: resvsca 29830 resvlem 29831

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