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Mathbox for Thierry Arnoux |
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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 11697 | . 2 ⊢ (1 − 1) = 0 | |
2 | simpl 486 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ LVec) | |
3 | simpr 488 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ DivRing) | |
4 | drngring 19502 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
5 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2798 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
7 | 5, 6 | ringidcl 19314 | . . . . . 6 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ (Base‘𝐹)) |
8 | 3, 4, 7 | 3syl 18 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1r‘𝐹) ∈ (Base‘𝐹)) |
9 | eqid 2798 | . . . . . . 7 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
10 | 9, 6 | drngunz 19510 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) ≠ (0g‘𝐹)) |
11 | 10 | adantl 485 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1r‘𝐹) ≠ (0g‘𝐹)) |
12 | eqid 2798 | . . . . . 6 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
13 | eqid 2798 | . . . . . 6 ⊢ (𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)})) = (𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)})) | |
14 | 5, 12, 9, 13 | lsatdim 31103 | . . . . 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 1368 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) = 1) |
16 | lveclmod 19871 | . . . . . . 7 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ LMod) | |
17 | 16 | adantr 484 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ LMod) |
18 | 8 | snssd 4702 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → {(1r‘𝐹)} ⊆ (Base‘𝐹)) |
19 | eqid 2798 | . . . . . . 7 ⊢ (LSubSp‘𝐹) = (LSubSp‘𝐹) | |
20 | 5, 19, 12 | lspcl 19741 | . . . . . 6 ⊢ ((𝐹 ∈ LMod ∧ {(1r‘𝐹)} ⊆ (Base‘𝐹)) → ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) |
21 | 17, 18, 20 | syl2anc 587 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) |
22 | 13 | lssdimle 31094 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) ≤ (dim‘𝐹)) |
23 | 2, 21, 22 | syl2anc 587 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) ≤ (dim‘𝐹)) |
24 | 15, 23 | eqbrtrrd 5054 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 1 ≤ (dim‘𝐹)) |
25 | 1nn0 11901 | . . . 4 ⊢ 1 ∈ ℕ0 | |
26 | dimcl 31091 | . . . . 5 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) ∈ ℕ0*) | |
27 | 26 | adantr 484 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘𝐹) ∈ ℕ0*) |
28 | xnn0lem1lt 12625 | . . . 4 ⊢ ((1 ∈ ℕ0 ∧ (dim‘𝐹) ∈ ℕ0*) → (1 ≤ (dim‘𝐹) ↔ (1 − 1) < (dim‘𝐹))) | |
29 | 25, 27, 28 | sylancr 590 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1 ≤ (dim‘𝐹) ↔ (1 − 1) < (dim‘𝐹))) |
30 | 24, 29 | mpbid 235 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1 − 1) < (dim‘𝐹)) |
31 | 1, 30 | eqbrtrrid 5066 | 1 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 < clt 10664 ≤ cle 10665 − cmin 10859 ℕ0cn0 11885 ℕ0*cxnn0 11955 Basecbs 16475 ↾s cress 16476 0gc0g 16705 1rcur 19244 Ringcrg 19290 DivRingcdr 19495 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LVecclvec 19867 dimcldim 31087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rpss 7429 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-r1 9177 df-rank 9178 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-ocomp 16578 df-0g 16707 df-mre 16849 df-mrc 16850 df-mri 16851 df-acs 16852 df-proset 17530 df-drs 17531 df-poset 17548 df-ipo 17754 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lbs 19840 df-lvec 19868 df-nzr 20024 df-lindf 20495 df-linds 20496 df-dim 31088 |
This theorem is referenced by: extdggt0 31135 |
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