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Mirrors > Home > HSE Home > Th. List > chne0 | Structured version Visualization version GIF version |
Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chne0 | ⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 3000 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ≠ 0ℋ ↔ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ≠ 0ℋ)) | |
2 | rexeq 3318 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ ↔ ∃𝑥 ∈ if (𝐴 ∈ Cℋ , 𝐴, 0ℋ)𝑥 ≠ 0ℎ)) | |
3 | 1, 2 | bibi12d 345 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ≠ 0ℋ ↔ ∃𝑥 ∈ if (𝐴 ∈ Cℋ , 𝐴, 0ℋ)𝑥 ≠ 0ℎ))) |
4 | h0elch 31064 | . . . 4 ⊢ 0ℋ ∈ Cℋ | |
5 | 4 | elimel 4598 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
6 | 5 | chne0i 31262 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ≠ 0ℋ ↔ ∃𝑥 ∈ if (𝐴 ∈ Cℋ , 𝐴, 0ℋ)𝑥 ≠ 0ℎ) |
7 | 3, 6 | dedth 4587 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 ifcif 4529 0ℎc0v 30733 Cℋ cch 30738 0ℋc0h 30744 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30808 ax-hfvadd 30809 ax-hvcom 30810 ax-hvass 30811 ax-hv0cl 30812 ax-hvaddid 30813 ax-hfvmul 30814 ax-hvmulid 30815 ax-hvmulass 30816 ax-hvdistr1 30817 ax-hvdistr2 30818 ax-hvmul0 30819 ax-hfi 30888 ax-his1 30891 ax-his2 30892 ax-his3 30893 ax-his4 30894 |
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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-icc 13363 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-topgen 17424 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-top 22795 df-topon 22812 df-bases 22848 df-lm 23132 df-haus 23218 df-grpo 30302 df-gid 30303 df-ginv 30304 df-gdiv 30305 df-ablo 30354 df-vc 30368 df-nv 30401 df-va 30404 df-ba 30405 df-sm 30406 df-0v 30407 df-vs 30408 df-nmcv 30409 df-ims 30410 df-hnorm 30777 df-hvsub 30780 df-hlim 30781 df-sh 31016 df-ch 31030 df-ch0 31062 |
This theorem is referenced by: atom1d 32162 |
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