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Theorem nvmf 30741

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 776	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
3		neg1cn 12142	. . . . . . 7 ⊢ -1 ∈ ℂ
4		nvmf.1	. . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈)
5		eqid 2740	. . . . . . . 8 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
6	4, 5	nvscl 30722	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
7	3, 6	mp3an2 1457	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
8	7	adantrl 722	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋)
9		eqid 2740	. . . . . 6 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
10	4, 9	nvgcl 30716	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1( ·_𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) → (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
11	1, 2, 8, 10	syl3anc 1379	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
12	11	ralrimivva 3183	. . 3 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋)
13		eqid 2740	. . . 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 8017	. . 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 219	. 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 30740	. . 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 258	1 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 × cxp 5623 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 ℂcc 11034 1c1 11037 -cneg 11376 NrmCVeccnv 30680 +_𝑣 cpv 30681 BaseSetcba 30682 ·_𝑠OLD cns 30683 −_𝑣 cnsb 30685

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-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112

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-nel 3040 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-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-id 5520 df-po 5533 df-so 5534 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-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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-ltxr 11182 df-sub 11377 df-neg 11378 df-grpo 30589 df-gid 30590 df-ginv 30591 df-gdiv 30592 df-ablo 30641 df-vc 30655 df-nv 30688 df-va 30691 df-ba 30692 df-sm 30693 df-0v 30694 df-vs 30695 df-nmcv 30696

This theorem is referenced by: nvmcl 30742 imsdval 30782 imsdf 30785 sspm 30830 hhssvsf 31369

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