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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 29028 | . . . . . 6 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐶 ∈ 𝐴) | |
2 | 1 | 3exp 1110 | . . . . 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 407 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ (span‘{𝐵})) → (𝐵 ∈ 𝐴 → 𝐶 ∈ 𝐴)) |
5 | 4 | ad2ant2r 743 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐵 ∈ 𝐴 → 𝐶 ∈ 𝐴)) |
6 | spansnid 29019 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
7 | 6 | ad2antrr 722 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐵 ∈ (span‘{𝐵})) |
8 | spansneleq 29026 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) → (𝐶 ∈ (span‘{𝐵}) → (span‘{𝐶}) = (span‘{𝐵}))) | |
9 | 8 | imp 407 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → (span‘{𝐶}) = (span‘{𝐵})) |
10 | 7, 9 | eleqtrrd 2884 | . . . . . . . 8 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ≠ 0ℎ) ∧ 𝐶 ∈ (span‘{𝐵})) → 𝐵 ∈ (span‘{𝐶})) |
11 | elspansn3 29028 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ (span‘{𝐶})) → 𝐵 ∈ 𝐴) | |
12 | 11 | 3expia 1112 | . . . . . . . 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 431 | . . . . . 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 432 | . . . 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 428 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ (span‘{𝐵}) ∧ 𝐶 ≠ 0ℎ)) → (𝐶 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
19 | 5, 18 | impbid 213 | 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 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 {csn 4466 ‘cfv 6217 ℋchba 28375 0ℎc0v 28380 Sℋ csh 28384 spancspn 28388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cc 9692 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452 ax-hilex 28455 ax-hfvadd 28456 ax-hvcom 28457 ax-hvass 28458 ax-hv0cl 28459 ax-hvaddid 28460 ax-hfvmul 28461 ax-hvmulid 28462 ax-hvmulass 28463 ax-hvdistr1 28464 ax-hvdistr2 28465 ax-hvmul0 28466 ax-hfi 28535 ax-his1 28538 ax-his2 28539 ax-his3 28540 ax-his4 28541 ax-hcompl 28658 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-omul 7949 df-er 8130 df-map 8249 df-pm 8250 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-card 9203 df-acn 9206 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ioo 12581 df-ico 12583 df-icc 12584 df-fz 12732 df-fzo 12873 df-fl 13000 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-rlim 14668 df-sum 14865 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-pt 16535 df-prds 16538 df-xrs 16592 df-qtop 16597 df-imas 16598 df-xps 16600 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-submnd 17763 df-mulg 17970 df-cntz 18176 df-cmn 18623 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-fbas 20212 df-fg 20213 df-cnfld 20216 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-cld 21299 df-ntr 21300 df-cls 21301 df-nei 21378 df-cn 21507 df-cnp 21508 df-lm 21509 df-haus 21595 df-tx 21842 df-hmeo 22035 df-fil 22126 df-fm 22218 df-flim 22219 df-flf 22220 df-xms 22601 df-ms 22602 df-tms 22603 df-cfil 23529 df-cau 23530 df-cmet 23531 df-grpo 27949 df-gid 27950 df-ginv 27951 df-gdiv 27952 df-ablo 28001 df-vc 28015 df-nv 28048 df-va 28051 df-ba 28052 df-sm 28053 df-0v 28054 df-vs 28055 df-nmcv 28056 df-ims 28057 df-dip 28157 df-ssp 28178 df-ph 28269 df-cbn 28319 df-hnorm 28424 df-hba 28425 df-hvsub 28427 df-hlim 28428 df-hcau 28429 df-sh 28663 df-ch 28677 df-oc 28708 df-ch0 28709 df-span 28765 |
This theorem is referenced by: elspansn5 29030 |
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