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Mirrors > Home > MPE Home > Th. List > nvpncan | Structured version Visualization version GIF version |
Description: Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvpncan2.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvpncan2.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvpncan2.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvpncan | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvpncan2.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nvpncan2.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
3 | 1, 2 | nvcom 28977 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐺𝐴) = (𝐴𝐺𝐵)) |
4 | 3 | oveq1d 7284 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐵𝐺𝐴)𝑀𝐵) = ((𝐴𝐺𝐵)𝑀𝐵)) |
5 | nvpncan2.3 | . . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
6 | 1, 2, 5 | nvpncan2 29009 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐵𝐺𝐴)𝑀𝐵) = 𝐴) |
7 | 4, 6 | eqtr3d 2782 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴) |
8 | 7 | 3com23 1125 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 NrmCVeccnv 28940 +𝑣 cpv 28941 BaseSetcba 28942 −𝑣 cnsb 28945 |
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 7580 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 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-nel 3052 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 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-1st 7822 df-2nd 7823 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-ltxr 11013 df-sub 11205 df-neg 11206 df-grpo 28849 df-gid 28850 df-ginv 28851 df-gdiv 28852 df-ablo 28901 df-vc 28915 df-nv 28948 df-va 28951 df-ba 28952 df-sm 28953 df-0v 28954 df-vs 28955 df-nmcv 28956 |
This theorem is referenced by: nvnpcan 29012 |
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