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Mirrors > Home > MPE Home > Th. List > nvm | Structured version Visualization version GIF version |
Description: Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvm.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvm.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvm.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
nvm.6 | ⊢ 𝑁 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
nvm | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvm.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | nvm.3 | . . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
3 | 1, 2 | vsfval 28039 | . . . 4 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
4 | nvm.6 | . . . 4 ⊢ 𝑁 = ( /𝑔 ‘𝐺) | |
5 | 3, 4 | eqtr4i 2852 | . . 3 ⊢ 𝑀 = 𝑁 |
6 | 5 | oveqi 6923 | . 2 ⊢ (𝐴𝑀𝐵) = (𝐴𝑁𝐵) |
7 | 6 | a1i 11 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 /𝑔 cgs 27898 NrmCVeccnv 27990 +𝑣 cpv 27991 BaseSetcba 27992 −𝑣 cnsb 27995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-grpo 27899 df-gdiv 27902 df-va 28001 df-vs 28005 |
This theorem is referenced by: nvmval 28048 |
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