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Theorem nvmf 29208

Description: Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvmf.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvmf.3	⊢ 𝑀 = ( −_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
nvmf	⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem nvmf Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑈 ∈ NrmCVec)
2		simprl 768	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
3		neg1cn 12180	. . . . . . 7 ⊢ -1 ∈ ℂ
4		nvmf.1	. . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈)
5		eqid 2736	. . . . . . . 8 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
6	4, 5	nvscl 29189	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
7	3, 6	mp3an2 1448	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
8	7	adantrl 713	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
9		eqid 2736	. . . . . 6 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
10	4, 9	nvgcl 29183	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) → (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
11	1, 2, 8, 10	syl3anc 1370	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
12	11	ralrimivva 3193	. . 3 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
13		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)))
14	13	fmpo 7968	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))):(𝑋 × 𝑋)⟶𝑋)
15	12, 14	sylib 217	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))):(𝑋 × 𝑋)⟶𝑋)
16		nvmf.3	. . . 4 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
17	4, 9, 5, 16	nvmfval 29207	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))))
18	17	feq1d 6630	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑀:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))):(𝑋 × 𝑋)⟶𝑋))
19	15, 18	mpbird 256	1 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 × cxp 5612 ⟶wf 6469 ‘cfv 6473 (class class class)co 7329 ∈ cmpo 7331 ℂcc 10962 1c1 10965 -cneg 11299 NrmCVeccnv 29147 +_𝑣 cpv 29148 BaseSetcba 29149 ·_𝑠OLD cns 29150 −_𝑣 cnsb 29152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-id 5512 df-po 5526 df-so 5527 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-1st 7891 df-2nd 7892 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-pnf 11104 df-mnf 11105 df-ltxr 11107 df-sub 11300 df-neg 11301 df-grpo 29056 df-gid 29057 df-ginv 29058 df-gdiv 29059 df-ablo 29108 df-vc 29122 df-nv 29155 df-va 29158 df-ba 29159 df-sm 29160 df-0v 29161 df-vs 29162 df-nmcv 29163

This theorem is referenced by: nvmcl 29209 imsdval 29249 imsdf 29252 sspm 29297 hhssvsf 29836

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