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 11707 | . 2 ⊢ (1 − 1) = 0 | |
2 | simpl 485 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ LVec) | |
3 | simpr 487 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ DivRing) | |
4 | drngring 19505 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
5 | eqid 2820 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
6 | eqid 2820 | . . . . . . 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 2820 | . . . . . . 7 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
10 | 9, 6 | drngunz 19513 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) ≠ (0g‘𝐹)) |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1r‘𝐹) ≠ (0g‘𝐹)) |
12 | eqid 2820 | . . . . . 6 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
13 | eqid 2820 | . . . . . 6 ⊢ (𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)})) = (𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)})) | |
14 | 5, 12, 9, 13 | lsatdim 31039 | . . . . 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 1366 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) = 1) |
16 | lveclmod 19874 | . . . . . . 7 ⊢ (𝐹 ∈ LVec → 𝐹 ∈ LMod) | |
17 | 16 | adantr 483 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 𝐹 ∈ LMod) |
18 | 8 | snssd 4739 | . . . . . 6 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → {(1r‘𝐹)} ⊆ (Base‘𝐹)) |
19 | eqid 2820 | . . . . . . 7 ⊢ (LSubSp‘𝐹) = (LSubSp‘𝐹) | |
20 | 5, 19, 12 | lspcl 19744 | . . . . . 6 ⊢ ((𝐹 ∈ LMod ∧ {(1r‘𝐹)} ⊆ (Base‘𝐹)) → ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) |
21 | 17, 18, 20 | syl2anc 586 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) |
22 | 13 | lssdimle 31030 | . . . . 5 ⊢ ((𝐹 ∈ LVec ∧ ((LSpan‘𝐹)‘{(1r‘𝐹)}) ∈ (LSubSp‘𝐹)) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) ≤ (dim‘𝐹)) |
23 | 2, 21, 22 | syl2anc 586 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘(𝐹 ↾s ((LSpan‘𝐹)‘{(1r‘𝐹)}))) ≤ (dim‘𝐹)) |
24 | 15, 23 | eqbrtrrd 5087 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 1 ≤ (dim‘𝐹)) |
25 | 1nn0 11911 | . . . 4 ⊢ 1 ∈ ℕ0 | |
26 | dimcl 31027 | . . . . 5 ⊢ (𝐹 ∈ LVec → (dim‘𝐹) ∈ ℕ0*) | |
27 | 26 | adantr 483 | . . . 4 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (dim‘𝐹) ∈ ℕ0*) |
28 | xnn0lem1lt 12635 | . . . 4 ⊢ ((1 ∈ ℕ0 ∧ (dim‘𝐹) ∈ ℕ0*) → (1 ≤ (dim‘𝐹) ↔ (1 − 1) < (dim‘𝐹))) | |
29 | 25, 27, 28 | sylancr 589 | . . 3 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1 ≤ (dim‘𝐹) ↔ (1 − 1) < (dim‘𝐹))) |
30 | 24, 29 | mpbid 234 | . 2 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → (1 − 1) < (dim‘𝐹)) |
31 | 1, 30 | eqbrtrrid 5099 | 1 ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ⊆ wss 3933 {csn 4564 class class class wbr 5063 ‘cfv 6352 (class class class)co 7153 0cc0 10534 1c1 10535 < clt 10672 ≤ cle 10673 − cmin 10867 ℕ0cn0 11895 ℕ0*cxnn0 11965 Basecbs 16479 ↾s cress 16480 0gc0g 16709 1rcur 19247 Ringcrg 19293 DivRingcdr 19498 LModclmod 19630 LSubSpclss 19699 LSpanclspn 19739 LVecclvec 19870 dimcldim 31023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-reg 9053 ax-inf2 9101 ax-ac2 9882 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-iin 4919 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-rpss 7446 df-om 7578 df-1st 7686 df-2nd 7687 df-tpos 7889 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-map 8405 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-oi 8971 df-r1 9190 df-rank 9191 df-dju 9327 df-card 9365 df-acn 9368 df-ac 9539 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-xnn0 11966 df-z 11980 df-dec 12097 df-uz 12242 df-fz 12891 df-hash 13689 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-tset 16580 df-ple 16581 df-ocomp 16582 df-0g 16711 df-mre 16853 df-mrc 16854 df-mri 16855 df-acs 16856 df-proset 17534 df-drs 17535 df-poset 17552 df-ipo 17758 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-submnd 17953 df-grp 18102 df-minusg 18103 df-sbg 18104 df-subg 18272 df-cmn 18904 df-abl 18905 df-mgp 19236 df-ur 19248 df-ring 19295 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19500 df-lmod 19632 df-lss 19700 df-lsp 19740 df-lbs 19843 df-lvec 19871 df-nzr 20027 df-lindf 20946 df-linds 20947 df-dim 31024 |
This theorem is referenced by: extdggt0 31071 |
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