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| Mirrors > Home > MPE Home > Th. List > nvnpcan | Structured version Visualization version GIF version | ||
| Description: Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvpncan2.1 | β’ π = (BaseSetβπ) |
| nvpncan2.2 | β’ πΊ = ( +π£ βπ) |
| nvpncan2.3 | β’ π = ( βπ£ βπ) |
| Ref | Expression |
|---|---|
| nvnpcan | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅)πΊπ΅) = π΄) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simprl 768 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β π΄ β π) | |
| 2 | simprr 770 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 3 | 1, 2, 2 | 3jca 1127 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β (π΄ β π β§ π΅ β π β§ π΅ β π)) |
| 4 | nvpncan2.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 5 | nvpncan2.2 | . . . . 5 β’ πΊ = ( +π£ βπ) | |
| 6 | nvpncan2.3 | . . . . 5 β’ π = ( βπ£ βπ) | |
| 7 | 4, 5, 6 | nvaddsub 30176 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ π΅ β π)) β ((π΄πΊπ΅)ππ΅) = ((π΄ππ΅)πΊπ΅)) |
| 8 | 3, 7 | syldan 590 | . . 3 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β ((π΄πΊπ΅)ππ΅) = ((π΄ππ΅)πΊπ΅)) |
| 9 | 8 | 3impb 1114 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄πΊπ΅)ππ΅) = ((π΄ππ΅)πΊπ΅)) |
| 10 | 4, 5, 6 | nvpncan 30175 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄πΊπ΅)ππ΅) = π΄) |
| 11 | 9, 10 | eqtr3d 2773 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅)πΊπ΅) = π΄) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 NrmCVeccnv 30105 +π£ cpv 30106 BaseSetcba 30107 βπ£ cnsb 30110 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-ltxr 11258 df-sub 11451 df-neg 11452 df-grpo 30014 df-gid 30015 df-ginv 30016 df-gdiv 30017 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-vs 30120 df-nmcv 30121 |
| This theorem is referenced by: nvmeq0 30179 |
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