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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 | β’ π = (BaseSetβπ) |
| nvmeq0.3 | β’ π = ( βπ£ βπ) |
| nvmeq0.5 | β’ π = (0vecβπ) |
| Ref | Expression |
|---|---|
| nvmeq0 | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅) = π β π΄ = π΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvmeq0.1 | . . . . . . 7 β’ π = (BaseSetβπ) | |
| 2 | nvmeq0.3 | . . . . . . 7 β’ π = ( βπ£ βπ) | |
| 3 | 1, 2 | nvmcl 30164 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) β π) |
| 4 | 3 | 3expb 1118 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β (π΄ππ΅) β π) |
| 5 | nvmeq0.5 | . . . . . . 7 β’ π = (0vecβπ) | |
| 6 | 1, 5 | nvzcl 30152 | . . . . . 6 β’ (π β NrmCVec β π β π) |
| 7 | 6 | adantr 479 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β π β π) |
| 8 | simprr 769 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 9 | 4, 7, 8 | 3jca 1126 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β ((π΄ππ΅) β π β§ π β π β§ π΅ β π)) |
| 10 | eqid 2730 | . . . . 5 β’ ( +π£ βπ) = ( +π£ βπ) | |
| 11 | 1, 10 | nvrcan 30142 | . . . 4 β’ ((π β NrmCVec β§ ((π΄ππ΅) β π β§ π β π β§ π΅ β π)) β (((π΄ππ΅)( +π£ βπ)π΅) = (π( +π£ βπ)π΅) β (π΄ππ΅) = π)) |
| 12 | 9, 11 | syldan 589 | . . 3 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β (((π΄ππ΅)( +π£ βπ)π΅) = (π( +π£ βπ)π΅) β (π΄ππ΅) = π)) |
| 13 | 12 | 3impb 1113 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (((π΄ππ΅)( +π£ βπ)π΅) = (π( +π£ βπ)π΅) β (π΄ππ΅) = π)) |
| 14 | 1, 10, 2 | nvnpcan 30174 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅)( +π£ βπ)π΅) = π΄) |
| 15 | 1, 10, 5 | nv0lid 30154 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π) β (π( +π£ βπ)π΅) = π΅) |
| 16 | 15 | 3adant2 1129 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π( +π£ βπ)π΅) = π΅) |
| 17 | 14, 16 | eqeq12d 2746 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (((π΄ππ΅)( +π£ βπ)π΅) = (π( +π£ βπ)π΅) β π΄ = π΅)) |
| 18 | 13, 17 | bitr3d 280 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅) = π β π΄ = π΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 βcfv 6544 (class class class)co 7413 NrmCVeccnv 30102 +π£ cpv 30103 BaseSetcba 30104 0veccn0v 30106 βπ£ cnsb 30107 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7979 df-2nd 7980 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-ltxr 11259 df-sub 11452 df-neg 11453 df-grpo 30011 df-gid 30012 df-ginv 30013 df-gdiv 30014 df-ablo 30063 df-vc 30077 df-nv 30110 df-va 30113 df-ba 30114 df-sm 30115 df-0v 30116 df-vs 30117 df-nmcv 30118 |
| This theorem is referenced by: nvmid 30177 ip2eqi 30374 |
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