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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rlmdim | Structured version Visualization version GIF version |
Description: The left vector space induced by a ring over itself has dimension 1. (Contributed by Thierry Arnoux, 5-Aug-2023.) Generalize to division rings. (Revised by SN, 22-Mar-2025.) |
Ref | Expression |
---|---|
rlmdim.1 | ⊢ 𝑉 = (ringLMod‘𝐹) |
Ref | Expression |
---|---|
rlmdim | ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmdim.1 | . . . 4 ⊢ 𝑉 = (ringLMod‘𝐹) | |
2 | rlmlvec 21229 | . . . 4 ⊢ (𝐹 ∈ DivRing → (ringLMod‘𝐹) ∈ LVec) | |
3 | 1, 2 | eqeltrid 2843 | . . 3 ⊢ (𝐹 ∈ DivRing → 𝑉 ∈ LVec) |
4 | ssid 4018 | . . . . . . . 8 ⊢ (Base‘𝐹) ⊆ (Base‘𝐹) | |
5 | rlmval 21216 | . . . . . . . . . 10 ⊢ (ringLMod‘𝐹) = ((subringAlg ‘𝐹)‘(Base‘𝐹)) | |
6 | 1, 5 | eqtri 2763 | . . . . . . . . 9 ⊢ 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹)) |
7 | eqid 2735 | . . . . . . . . 9 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
8 | 6, 7 | sradrng 33613 | . . . . . . . 8 ⊢ ((𝐹 ∈ DivRing ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ DivRing) |
9 | 4, 8 | mpan2 691 | . . . . . . 7 ⊢ (𝐹 ∈ DivRing → 𝑉 ∈ DivRing) |
10 | 9 | drngringd 20754 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝑉 ∈ Ring) |
11 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
12 | eqid 2735 | . . . . . . 7 ⊢ (1r‘𝑉) = (1r‘𝑉) | |
13 | 11, 12 | ringidcl 20280 | . . . . . 6 ⊢ (𝑉 ∈ Ring → (1r‘𝑉) ∈ (Base‘𝑉)) |
14 | 10, 13 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ DivRing → (1r‘𝑉) ∈ (Base‘𝑉)) |
15 | eqid 2735 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
16 | 15, 12 | drngunz 20764 | . . . . . 6 ⊢ (𝑉 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
17 | 9, 16 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
18 | 11, 15 | lindssn 33386 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (1r‘𝑉) ∈ (Base‘𝑉) ∧ (1r‘𝑉) ≠ (0g‘𝑉)) → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
19 | 3, 14, 17, 18 | syl3anc 1370 | . . . 4 ⊢ (𝐹 ∈ DivRing → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
20 | drngring 20753 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
21 | 1 | fveq2i 6910 | . . . . . . . 8 ⊢ (LSpan‘𝑉) = (LSpan‘(ringLMod‘𝐹)) |
22 | rspval 21239 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (LSpan‘(ringLMod‘𝐹)) | |
23 | 21, 22 | eqtr4i 2766 | . . . . . . 7 ⊢ (LSpan‘𝑉) = (RSpan‘𝐹) |
24 | eqid 2735 | . . . . . . 7 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
25 | 23, 7, 24 | rsp1 21265 | . . . . . 6 ⊢ (𝐹 ∈ Ring → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
26 | 20, 25 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ DivRing → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
27 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing → 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹))) |
28 | eqidd 2736 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) = (1r‘𝐹)) | |
29 | ssidd 4019 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (Base‘𝐹) ⊆ (Base‘𝐹)) | |
30 | 27, 28, 29 | sra1r 33612 | . . . . . . 7 ⊢ (𝐹 ∈ DivRing → (1r‘𝐹) = (1r‘𝑉)) |
31 | 30 | sneqd 4643 | . . . . . 6 ⊢ (𝐹 ∈ DivRing → {(1r‘𝐹)} = {(1r‘𝑉)}) |
32 | 31 | fveq2d 6911 | . . . . 5 ⊢ (𝐹 ∈ DivRing → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝑉)})) |
33 | 27, 29 | srabase 21195 | . . . . 5 ⊢ (𝐹 ∈ DivRing → (Base‘𝐹) = (Base‘𝑉)) |
34 | 26, 32, 33 | 3eqtr3d 2783 | . . . 4 ⊢ (𝐹 ∈ DivRing → ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉)) |
35 | eqid 2735 | . . . . 5 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉) | |
36 | eqid 2735 | . . . . 5 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉) | |
37 | 11, 35, 36 | islbs4 21870 | . . . 4 ⊢ ({(1r‘𝑉)} ∈ (LBasis‘𝑉) ↔ ({(1r‘𝑉)} ∈ (LIndS‘𝑉) ∧ ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉))) |
38 | 19, 34, 37 | sylanbrc 583 | . . 3 ⊢ (𝐹 ∈ DivRing → {(1r‘𝑉)} ∈ (LBasis‘𝑉)) |
39 | 35 | dimval 33628 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ {(1r‘𝑉)} ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
40 | 3, 38, 39 | syl2anc 584 | . 2 ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
41 | fvex 6920 | . . 3 ⊢ (1r‘𝑉) ∈ V | |
42 | hashsng 14405 | . . 3 ⊢ ((1r‘𝑉) ∈ V → (♯‘{(1r‘𝑉)}) = 1) | |
43 | 41, 42 | ax-mp 5 | . 2 ⊢ (♯‘{(1r‘𝑉)}) = 1 |
44 | 40, 43 | eqtrdi 2791 | 1 ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 {csn 4631 ‘cfv 6563 1c1 11154 ♯chash 14366 Basecbs 17245 0gc0g 17486 1rcur 20199 Ringcrg 20251 DivRingcdr 20746 LSpanclspn 20987 LBasisclbs 21091 LVecclvec 21119 subringAlg csra 21188 ringLModcrglmod 21189 RSpancrsp 21235 LIndSclinds 21843 dimcldim 33626 |
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-reg 9630 ax-inf2 9679 ax-ac2 10501 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-iin 4999 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-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-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-r1 9802 df-rank 9803 df-card 9977 df-acn 9980 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-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ocomp 17319 df-0g 17488 df-mre 17631 df-mrc 17632 df-mri 17633 df-acs 17634 df-proset 18352 df-drs 18353 df-poset 18371 df-ipo 18586 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 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-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lbs 21092 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-lindf 21844 df-linds 21845 df-dim 33627 |
This theorem is referenced by: extdgid 33688 |
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