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Mirrors > Home > MPE Home > Th. List > lbsexg | Structured version Visualization version GIF version |
Description: Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
lbsex.j | ⊢ 𝐽 = (LBasis‘𝑊) |
Ref | Expression |
---|---|
lbsexg | ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → 𝐽 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LVec) | |
2 | fvex 6901 | . . . . 5 ⊢ (Base‘𝑊) ∈ V | |
3 | 2 | pwex 5377 | . . . 4 ⊢ 𝒫 (Base‘𝑊) ∈ V |
4 | dfac10 10128 | . . . . 5 ⊢ (CHOICE ↔ dom card = V) | |
5 | 4 | biimpi 215 | . . . 4 ⊢ (CHOICE → dom card = V) |
6 | 3, 5 | eleqtrrid 2841 | . . 3 ⊢ (CHOICE → 𝒫 (Base‘𝑊) ∈ dom card) |
7 | 0ss 4395 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑊) | |
8 | ral0 4511 | . . . 4 ⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(∅ ∖ {𝑥})) | |
9 | lbsex.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
10 | eqid 2733 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
11 | eqid 2733 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
12 | 9, 10, 11 | lbsextg 20763 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝒫 (Base‘𝑊) ∈ dom card) ∧ ∅ ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(∅ ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
13 | 7, 8, 12 | mp3an23 1454 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝒫 (Base‘𝑊) ∈ dom card) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
14 | 1, 6, 13 | syl2anr 598 | . 2 ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
15 | rexn0 4509 | . 2 ⊢ (∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠 → 𝐽 ≠ ∅) | |
16 | 14, 15 | syl 17 | 1 ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → 𝐽 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 dom cdm 5675 ‘cfv 6540 cardccrd 9926 CHOICEwac 10106 Basecbs 17140 LSpanclspn 20570 LBasisclbs 20673 LVecclvec 20701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-rpss 7708 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-drng 20306 df-lmod 20461 df-lss 20531 df-lsp 20571 df-lbs 20674 df-lvec 20702 |
This theorem is referenced by: lbsex 20766 |
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