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Mirrors > Home > MPE Home > Th. List > nvnnncan1 | Structured version Visualization version GIF version |
Description: Cancellation law for vector subtraction. (nnncan1 11257 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvmf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvmf.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvnnncan1 | ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | 1 | nvablo 28974 | . 2 ⊢ (𝑈 ∈ NrmCVec → ( +𝑣 ‘𝑈) ∈ AbelOp) |
3 | nvmf.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 28962 | . . 3 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
5 | nvmf.3 | . . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
6 | 1, 5 | vsfval 28991 | . . 3 ⊢ 𝑀 = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
7 | 4, 6 | ablonnncan1 28915 | . 2 ⊢ ((( +𝑣 ‘𝑈) ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵)) |
8 | 2, 7 | sylan 580 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 AbelOpcablo 28902 NrmCVeccnv 28942 +𝑣 cpv 28943 BaseSetcba 28944 −𝑣 cnsb 28947 |
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 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-vs 28957 df-nmcv 28958 |
This theorem is referenced by: minvecolem2 29233 |
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