![]() |
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 6920 | . . . . 5 ⊢ (Base‘𝑊) ∈ V | |
3 | 2 | pwex 5386 | . . . 4 ⊢ 𝒫 (Base‘𝑊) ∈ V |
4 | dfac10 10176 | . . . . 5 ⊢ (CHOICE ↔ dom card = V) | |
5 | 4 | biimpi 216 | . . . 4 ⊢ (CHOICE → dom card = V) |
6 | 3, 5 | eleqtrrid 2846 | . . 3 ⊢ (CHOICE → 𝒫 (Base‘𝑊) ∈ dom card) |
7 | 0ss 4406 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑊) | |
8 | ral0 4519 | . . . 4 ⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(∅ ∖ {𝑥})) | |
9 | lbsex.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
10 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
11 | eqid 2735 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
12 | 9, 10, 11 | lbsextg 21182 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ 𝒫 (Base‘𝑊) ∈ dom card) ∧ ∅ ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ ((LSpan‘𝑊)‘(∅ ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
13 | 7, 8, 12 | mp3an23 1452 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝒫 (Base‘𝑊) ∈ dom card) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
14 | 1, 6, 13 | syl2anr 597 | . 2 ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → ∃𝑠 ∈ 𝐽 ∅ ⊆ 𝑠) |
15 | rexn0 4517 | . 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 dom cdm 5689 ‘cfv 6563 cardccrd 9973 CHOICEwac 10153 Basecbs 17245 LSpanclspn 20987 LBasisclbs 21091 LVecclvec 21119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7742 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lbs 21092 df-lvec 21120 |
This theorem is referenced by: lbsex 21185 |
Copyright terms: Public domain | W3C validator |