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Theorem nvmf 30720

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 482	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑈 ∈ NrmCVec)
2		simprl 770	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
3		neg1cn 12130	. . . . . . 7 ⊢ -1 ∈ ℂ
4		nvmf.1	. . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈)
5		eqid 2736	. . . . . . . 8 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
6	4, 5	nvscl 30701	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
7	3, 6	mp3an2 1451	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
8	7	adantrl 716	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
9		eqid 2736	. . . . . 6 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
10	4, 9	nvgcl 30695	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) → (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
11	1, 2, 8, 10	syl3anc 1373	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
12	11	ralrimivva 3179	. . 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 8012	. . 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 218	. 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 30719	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))))
18	17	feq1d 6644	. 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 257	1 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 × cxp 5622 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ℂcc 11024 1c1 11027 -cneg 11365 NrmCVeccnv 30659 +_𝑣 cpv 30660 BaseSetcba 30661 ·_𝑠OLD cns 30662 −_𝑣 cnsb 30664

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 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-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-ltxr 11171 df-sub 11366 df-neg 11367 df-grpo 30568 df-gid 30569 df-ginv 30570 df-gdiv 30571 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-vs 30674 df-nmcv 30675

This theorem is referenced by: nvmcl 30721 imsdval 30761 imsdf 30764 sspm 30809 hhssvsf 31348

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