![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > drngdimgt0 | Structured version Visualization version GIF version |
Description: The dimension of a vector space that is also a division ring is greater than zero. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
drngdimgt0 | ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1m1e0 12271 | . 2 ⊢ (1 − 1) = 0 | |
2 | simpl 484 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ LVec) | |
3 | simpr 486 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ DivRing) | |
4 | drngring 20300 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
5 | eqid 2733 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2733 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
7 | 5, 6 | ringidcl 20064 | . . . . . 6 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ (Base‘𝐹)) |
8 | 3, 4, 7 | 3syl 18 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1r‘𝐹) ∈ (Base‘𝐹)) |
9 | eqid 2733 | . . . . . . 7 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
10 | 9, 6 | drngunz 20311 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) ≠ (0g‘𝐹)) |
11 | 10 | adantl 483 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1r‘𝐹) ≠ (0g‘𝐹)) |
12 | eqid 2733 | . . . . . 6 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
13 | eqid 2733 | . . . . . 6 ⊢ (𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)})) = (𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)})) | |
14 | 5, 12, 9, 13 | lsatdim 32640 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ (1r‘𝐹) ∈ (Base‘𝐹) ∧ (1r‘𝐹) ≠ (0g‘𝐹)) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) = 1) |
15 | 2, 8, 11, 14 | syl3anc 1372 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) = 1) |
16 | lveclmod 20694 | . . . . . . 7 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ LMod) | |
17 | 16 | adantr 482 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ LMod) |
18 | 8 | snssd 4808 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → {(1r‘𝐹)} ⊆ (Base‘𝐹)) |
19 | eqid 2733 | . . . . . . 7 ⊢ (LSubSp‘𝐹) = (LSubSp‘𝐹) | |
20 | 5, 19, 12 | lspcl 20564 | . . . . . 6 ⊢ ((𝐹 ∈ LMod ∧ {(1r‘𝐹)} ⊆ (Base‘𝐹)) → ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) |
21 | 17, 18, 20 | syl2anc 585 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) |
22 | 13 | lssdimle 32631 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) ≤ (dim‘𝐹)) |
23 | 2, 21, 22 | syl2anc 585 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) ≤ (dim‘𝐹)) |
24 | 15, 23 | eqbrtrrd 5168 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 1 ≤ (dim‘𝐹)) |
25 | 1nn0 12475 | . . . 4 ⊢ 1 ∈ ℕ0 | |
26 | dimcl 32627 | . . . . 5 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) ∈ ℕ0*) | |
27 | 26 | adantr 482 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘𝐹) ∈ ℕ0*) |
28 | xnn0lem1lt 13210 | . . . 4 ⊢ ((1 ∈ ℕ0 ∧ (dim‘𝐹) ∈ ℕ0*) → (1 ≤ (dim‘𝐹) ↔ (1 − 1) < (dim‘𝐹))) | |
29 | 25, 27, 28 | sylancr 588 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1 ≤ (dim‘𝐹) ↔ (1 − 1) < (dim‘𝐹))) |
30 | 24, 29 | mpbid 231 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1 − 1) < (dim‘𝐹)) |
31 | 1, 30 | eqbrtrrid 5180 | 1 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3946 {csn 4624 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 0cc0 11097 1c1 11098 < clt 11235 ≤ cle 11236 − cmin 11431 ℕ0cn0 12459 ℕ0*cxnn0 12531 Basecbs 17131 ↾s cress 17160 0gc0g 17372 1rcur 19987 Ringcrg 20038 DivRingcdr 20293 LModclmod 20448 LSubSpclss 20519 LSpanclspn 20559 LVecclvec 20690 dimcldim 32623 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-reg 9574 ax-inf2 9623 ax-ac2 10445 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-rpss 7700 df-om 7843 df-1st 7962 df-2nd 7963 df-tpos 8198 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-oi 9492 df-r1 9746 df-rank 9747 df-dju 9883 df-card 9921 df-acn 9924 df-ac 10098 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-xnn0 12532 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-hash 14278 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-tset 17203 df-ple 17204 df-ocomp 17205 df-0g 17374 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18235 df-drs 18236 df-poset 18253 df-ipo 18468 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-nzr 20270 df-drng 20295 df-lmod 20450 df-lss 20520 df-lsp 20560 df-lbs 20663 df-lvec 20691 df-lindf 21334 df-linds 21335 df-dim 32624 |
This theorem is referenced by: extdggt0 32674 |
Copyright terms: Public domain | W3C validator |