Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6676 | . . . . 5 ⊢ (Base‘𝑊) ∈ V | |
3 | 2 | pwex 5272 | . . . 4 ⊢ 𝒫 (Base‘𝑊) ∈ V |
4 | dfac10 9551 | . . . . 5 ⊢ (CHOICE ↔ dom card = V) | |
5 | 4 | biimpi 217 | . . . 4 ⊢ (CHOICE → dom card = V) |
6 | 3, 5 | eleqtrrid 2917 | . . 3 ⊢ (CHOICE → 𝒫 (Base‘𝑊) ∈ dom card) |
7 | 0ss 4347 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑊) | |
8 | ral0 4452 | . . . 4 ⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(∅ ∖ {𝑥})) | |
9 | lbsex.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
10 | eqid 2818 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
11 | eqid 2818 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
12 | 9, 10, 11 | lbsextg 19863 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝒫 (Base‘𝑊) ∈ dom card) ∧ ∅ ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(∅ ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
13 | 7, 8, 12 | mp3an23 1444 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝒫 (Base‘𝑊) ∈ dom card) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
14 | 1, 6, 13 | syl2anr 596 | . 2 ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
15 | rexn0 4450 | . 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 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 {csn 4557 dom cdm 5548 ‘cfv 6348 cardccrd 9352 CHOICEwac 9529 Basecbs 16471 LSpanclspn 19672 LBasisclbs 19775 LVecclvec 19803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-rpss 7438 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lbs 19776 df-lvec 19804 |
This theorem is referenced by: lbsex 19866 |
Copyright terms: Public domain | W3C validator |