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Mirrors > Home > MPE Home > Th. List > sspz | Structured version Visualization version GIF version |
Description: The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspz.z | β’ π = (0vecβπ) |
sspz.q | β’ π = (0vecβπ) |
sspz.h | β’ π» = (SubSpβπ) |
Ref | Expression |
---|---|
sspz | β’ ((π β NrmCVec β§ π β π») β π = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspz.h | . . . . 5 β’ π» = (SubSpβπ) | |
2 | 1 | sspnv 30529 | . . . 4 β’ ((π β NrmCVec β§ π β π») β π β NrmCVec) |
3 | eqid 2728 | . . . . . 6 β’ (BaseSetβπ) = (BaseSetβπ) | |
4 | sspz.q | . . . . . 6 β’ π = (0vecβπ) | |
5 | 3, 4 | nvzcl 30437 | . . . . 5 β’ (π β NrmCVec β π β (BaseSetβπ)) |
6 | 5, 5 | jca 511 | . . . 4 β’ (π β NrmCVec β (π β (BaseSetβπ) β§ π β (BaseSetβπ))) |
7 | 2, 6 | syl 17 | . . 3 β’ ((π β NrmCVec β§ π β π») β (π β (BaseSetβπ) β§ π β (BaseSetβπ))) |
8 | eqid 2728 | . . . 4 β’ ( βπ£ βπ) = ( βπ£ βπ) | |
9 | eqid 2728 | . . . 4 β’ ( βπ£ βπ) = ( βπ£ βπ) | |
10 | 3, 8, 9, 1 | sspmval 30536 | . . 3 β’ (((π β NrmCVec β§ π β π») β§ (π β (BaseSetβπ) β§ π β (BaseSetβπ))) β (π( βπ£ βπ)π) = (π( βπ£ βπ)π)) |
11 | 7, 10 | mpdan 686 | . 2 β’ ((π β NrmCVec β§ π β π») β (π( βπ£ βπ)π) = (π( βπ£ βπ)π)) |
12 | 3, 9, 4 | nvmid 30462 | . . 3 β’ ((π β NrmCVec β§ π β (BaseSetβπ)) β (π( βπ£ βπ)π) = π) |
13 | 2, 5, 12 | syl2anc2 584 | . 2 β’ ((π β NrmCVec β§ π β π») β (π( βπ£ βπ)π) = π) |
14 | eqid 2728 | . . . . 5 β’ (BaseSetβπ) = (BaseSetβπ) | |
15 | 14, 3, 1 | sspba 30530 | . . . 4 β’ ((π β NrmCVec β§ π β π») β (BaseSetβπ) β (BaseSetβπ)) |
16 | 2, 5 | syl 17 | . . . 4 β’ ((π β NrmCVec β§ π β π») β π β (BaseSetβπ)) |
17 | 15, 16 | sseldd 3979 | . . 3 β’ ((π β NrmCVec β§ π β π») β π β (BaseSetβπ)) |
18 | sspz.z | . . . 4 β’ π = (0vecβπ) | |
19 | 14, 8, 18 | nvmid 30462 | . . 3 β’ ((π β NrmCVec β§ π β (BaseSetβπ)) β (π( βπ£ βπ)π) = π) |
20 | 17, 19 | syldan 590 | . 2 β’ ((π β NrmCVec β§ π β π») β (π( βπ£ βπ)π) = π) |
21 | 11, 13, 20 | 3eqtr3d 2776 | 1 β’ ((π β NrmCVec β§ π β π») β π = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 NrmCVeccnv 30387 BaseSetcba 30389 0veccn0v 30391 βπ£ cnsb 30392 SubSpcss 30524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470 df-neg 11471 df-grpo 30296 df-gid 30297 df-ginv 30298 df-gdiv 30299 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-vs 30402 df-nmcv 30403 df-ssp 30525 |
This theorem is referenced by: hhshsslem2 31071 |
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