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Mirrors > Home > HSE Home > Th. List > elspansn4 | Structured version Visualization version GIF version |
Description: A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elspansn4 | ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elspansn3 29510 | . . . . . 6 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐶 ∈ 𝐴) | |
2 | 1 | 3exp 1120 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐵 ∈ 𝐴 → (𝐶 ∈ (span‘{𝐵}) → 𝐶 ∈ 𝐴))) |
3 | 2 | com23 86 | . . . 4 ⊢ (𝐴 ∈ Sℋ → (𝐶 ∈ (span‘{𝐵}) → (𝐵 ∈ 𝐴 → 𝐶 ∈ 𝐴))) |
4 | 3 | imp 410 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ (span‘{𝐵})) → (𝐵 ∈ 𝐴 → 𝐶 ∈ 𝐴)) |
5 | 4 | ad2ant2r 747 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐵 ∈ 𝐴 → 𝐶 ∈ 𝐴)) |
6 | spansnid 29501 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
7 | 6 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐵 ∈ (span‘{𝐵})) |
8 | spansneleq 29508 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) → (𝐶 ∈ (span‘{𝐵}) → (span‘{𝐶}) = (span‘{𝐵}))) | |
9 | 8 | imp 410 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → (span‘{𝐶}) = (span‘{𝐵})) |
10 | 7, 9 | eleqtrrd 2837 | . . . . . . . 8 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐵 ∈ (span‘{𝐶})) |
11 | elspansn3 29510 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ (span‘{𝐶})) → 𝐵 ∈ 𝐴) | |
12 | 11 | 3expia 1122 | . . . . . . . 8 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ 𝐴) → (𝐵 ∈ (span‘{𝐶}) → 𝐵 ∈ 𝐴)) |
13 | 10, 12 | syl5 34 | . . . . . . 7 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ 𝐴) → (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐵 ∈ 𝐴)) |
14 | 13 | exp4b 434 | . . . . . 6 ⊢ (𝐴 ∈ Sℋ → (𝐶 ∈ 𝐴 → ((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) → (𝐶 ∈ (span‘{𝐵}) → 𝐵 ∈ 𝐴)))) |
15 | 14 | com24 95 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐶 ∈ (span‘{𝐵}) → ((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) → (𝐶 ∈ 𝐴 → 𝐵 ∈ 𝐴)))) |
16 | 15 | exp4a 435 | . . . 4 ⊢ (𝐴 ∈ Sℋ → (𝐶 ∈ (span‘{𝐵}) → (𝐵 ∈ ℋ → (𝐶 ≠ 0ℎ → (𝐶 ∈ 𝐴 → 𝐵 ∈ 𝐴))))) |
17 | 16 | com23 86 | . . 3 ⊢ (𝐴 ∈ Sℋ → (𝐵 ∈ ℋ → (𝐶 ∈ (span‘{𝐵}) → (𝐶 ≠ 0ℎ → (𝐶 ∈ 𝐴 → 𝐵 ∈ 𝐴))))) |
18 | 17 | imp43 431 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐶 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
19 | 5, 18 | impbid 215 | 1 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 {csn 4517 ‘cfv 6340 ℋchba 28857 0ℎc0v 28862 Sℋ csh 28866 spancspn 28870 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cc 9938 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 ax-mulf 10698 ax-hilex 28937 ax-hfvadd 28938 ax-hvcom 28939 ax-hvass 28940 ax-hv0cl 28941 ax-hvaddid 28942 ax-hfvmul 28943 ax-hvmulid 28944 ax-hvmulass 28945 ax-hvdistr1 28946 ax-hvdistr2 28947 ax-hvmul0 28948 ax-hfi 29017 ax-his1 29020 ax-his2 29021 ax-his3 29022 ax-his4 29023 ax-hcompl 29140 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-oadd 8138 df-omul 8139 df-er 8323 df-map 8442 df-pm 8443 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-fi 8951 df-sup 8982 df-inf 8983 df-oi 9050 df-card 9444 df-acn 9447 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-q 12434 df-rp 12476 df-xneg 12593 df-xadd 12594 df-xmul 12595 df-ioo 12828 df-ico 12830 df-icc 12831 df-fz 12985 df-fzo 13128 df-fl 13256 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-rlim 14939 df-sum 15139 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-hom 16695 df-cco 16696 df-rest 16802 df-topn 16803 df-0g 16821 df-gsum 16822 df-topgen 16823 df-pt 16824 df-prds 16827 df-xrs 16881 df-qtop 16886 df-imas 16887 df-xps 16889 df-mre 16963 df-mrc 16964 df-acs 16966 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-mulg 18346 df-cntz 18568 df-cmn 19029 df-psmet 20212 df-xmet 20213 df-met 20214 df-bl 20215 df-mopn 20216 df-fbas 20217 df-fg 20218 df-cnfld 20221 df-top 21648 df-topon 21665 df-topsp 21687 df-bases 21700 df-cld 21773 df-ntr 21774 df-cls 21775 df-nei 21852 df-cn 21981 df-cnp 21982 df-lm 21983 df-haus 22069 df-tx 22316 df-hmeo 22509 df-fil 22600 df-fm 22692 df-flim 22693 df-flf 22694 df-xms 23076 df-ms 23077 df-tms 23078 df-cfil 24010 df-cau 24011 df-cmet 24012 df-grpo 28431 df-gid 28432 df-ginv 28433 df-gdiv 28434 df-ablo 28483 df-vc 28497 df-nv 28530 df-va 28533 df-ba 28534 df-sm 28535 df-0v 28536 df-vs 28537 df-nmcv 28538 df-ims 28539 df-dip 28639 df-ssp 28660 df-ph 28751 df-cbn 28801 df-hnorm 28906 df-hba 28907 df-hvsub 28909 df-hlim 28910 df-hcau 28911 df-sh 29145 df-ch 29159 df-oc 29190 df-ch0 29191 df-span 29247 |
This theorem is referenced by: elspansn5 29512 |
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