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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 | β’ π = (BaseSetβπ) |
| nvpncan2.2 | β’ πΊ = ( +π£ βπ) |
| nvpncan2.3 | β’ π = ( βπ£ βπ) |
| Ref | Expression |
|---|---|
| nvpncan | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄πΊπ΅)ππ΅) = π΄) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvpncan2.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 2 | nvpncan2.2 | . . . . 5 β’ πΊ = ( +π£ βπ) | |
| 3 | 1, 2 | nvcom 30378 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π β§ π΄ β π) β (π΅πΊπ΄) = (π΄πΊπ΅)) |
| 4 | 3 | oveq1d 7419 | . . 3 β’ ((π β NrmCVec β§ π΅ β π β§ π΄ β π) β ((π΅πΊπ΄)ππ΅) = ((π΄πΊπ΅)ππ΅)) |
| 5 | nvpncan2.3 | . . . 4 β’ π = ( βπ£ βπ) | |
| 6 | 1, 2, 5 | nvpncan2 30410 | . . 3 β’ ((π β NrmCVec β§ π΅ β π β§ π΄ β π) β ((π΅πΊπ΄)ππ΅) = π΄) |
| 7 | 4, 6 | eqtr3d 2768 | . 2 β’ ((π β NrmCVec β§ π΅ β π β§ π΄ β π) β ((π΄πΊπ΅)ππ΅) = π΄) |
| 8 | 7 | 3com23 1123 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄πΊπ΅)ππ΅) = π΄) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 NrmCVeccnv 30341 +π£ cpv 30342 BaseSetcba 30343 βπ£ cnsb 30346 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-sub 11447 df-neg 11448 df-grpo 30250 df-gid 30251 df-ginv 30252 df-gdiv 30253 df-ablo 30302 df-vc 30316 df-nv 30349 df-va 30352 df-ba 30353 df-sm 30354 df-0v 30355 df-vs 30356 df-nmcv 30357 |
| This theorem is referenced by: nvnpcan 30413 |
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