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Mirrors > Home > MPE Home > Th. List > nvmeq0 | Structured version Visualization version GIF version |
Description: The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvmeq0.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvmeq0.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
nvmeq0.5 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
nvmeq0 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵) = 𝑍 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvmeq0.1 | . . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nvmeq0.3 | . . . . . . 7 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
3 | 1, 2 | nvmcl 28573 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) ∈ 𝑋) |
4 | 3 | 3expb 1121 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑀𝐵) ∈ 𝑋) |
5 | nvmeq0.5 | . . . . . . 7 ⊢ 𝑍 = (0vec‘𝑈) | |
6 | 1, 5 | nvzcl 28561 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑍 ∈ 𝑋) |
8 | simprr 773 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
9 | 4, 7, 8 | 3jca 1129 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝑀𝐵) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) |
10 | eqid 2738 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
11 | 1, 10 | nvrcan 28551 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ ((𝐴𝑀𝐵) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝐴𝑀𝐵)( +𝑣 ‘𝑈)𝐵) = (𝑍( +𝑣 ‘𝑈)𝐵) ↔ (𝐴𝑀𝐵) = 𝑍)) |
12 | 9, 11 | syldan 594 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝐴𝑀𝐵)( +𝑣 ‘𝑈)𝐵) = (𝑍( +𝑣 ‘𝑈)𝐵) ↔ (𝐴𝑀𝐵) = 𝑍)) |
13 | 12 | 3impb 1116 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴𝑀𝐵)( +𝑣 ‘𝑈)𝐵) = (𝑍( +𝑣 ‘𝑈)𝐵) ↔ (𝐴𝑀𝐵) = 𝑍)) |
14 | 1, 10, 2 | nvnpcan 28583 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵)( +𝑣 ‘𝑈)𝐵) = 𝐴) |
15 | 1, 10, 5 | nv0lid 28563 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑍( +𝑣 ‘𝑈)𝐵) = 𝐵) |
16 | 15 | 3adant2 1132 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑍( +𝑣 ‘𝑈)𝐵) = 𝐵) |
17 | 14, 16 | eqeq12d 2754 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴𝑀𝐵)( +𝑣 ‘𝑈)𝐵) = (𝑍( +𝑣 ‘𝑈)𝐵) ↔ 𝐴 = 𝐵)) |
18 | 13, 17 | bitr3d 284 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵) = 𝑍 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 NrmCVeccnv 28511 +𝑣 cpv 28512 BaseSetcba 28513 0veccn0v 28515 −𝑣 cnsb 28516 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-ltxr 10751 df-sub 10943 df-neg 10944 df-grpo 28420 df-gid 28421 df-ginv 28422 df-gdiv 28423 df-ablo 28472 df-vc 28486 df-nv 28519 df-va 28522 df-ba 28523 df-sm 28524 df-0v 28525 df-vs 28526 df-nmcv 28527 |
This theorem is referenced by: nvmid 28586 ip2eqi 28783 |
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