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Mirrors > Home > MPE Home > Th. List > Mathboxes > rgmoddimOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rlmdim 33150 as of 21-Mar-2025. (Contributed by Thierry Arnoux, 5-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rlmdim.1 | ⊢ 𝑉 = (ringLMod‘𝐹) |
Ref | Expression |
---|---|
rgmoddimOLD | ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfld 20594 | . . . . 5 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing)) | |
2 | 1 | simplbi 497 | . . . 4 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing) |
3 | eqid 2731 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | 3 | ressid 17196 | . . . . 5 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) = 𝐹) |
5 | 4, 2 | eqeltrd 2832 | . . . 4 ⊢ (𝐹 ∈ Field → (𝐹 ↾s (Base‘𝐹)) ∈ DivRing) |
6 | drngring 20590 | . . . . 5 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring) | |
7 | 3 | subrgid 20471 | . . . . 5 ⊢ (𝐹 ∈ Ring → (Base‘𝐹) ∈ (SubRing‘𝐹)) |
8 | 2, 6, 7 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ Field → (Base‘𝐹) ∈ (SubRing‘𝐹)) |
9 | rlmdim.1 | . . . . . 6 ⊢ 𝑉 = (ringLMod‘𝐹) | |
10 | rlmval 21043 | . . . . . 6 ⊢ (ringLMod‘𝐹) = ((subringAlg ‘𝐹)‘(Base‘𝐹)) | |
11 | 9, 10 | eqtri 2759 | . . . . 5 ⊢ 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹)) |
12 | eqid 2731 | . . . . 5 ⊢ (𝐹 ↾s (Base‘𝐹)) = (𝐹 ↾s (Base‘𝐹)) | |
13 | 11, 12 | sralvec 33128 | . . . 4 ⊢ ((𝐹 ∈ DivRing ∧ (𝐹 ↾s (Base‘𝐹)) ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐹)) → 𝑉 ∈ LVec) |
14 | 2, 5, 8, 13 | syl3anc 1370 | . . 3 ⊢ (𝐹 ∈ Field → 𝑉 ∈ LVec) |
15 | 2, 6 | syl 17 | . . . . . . 7 ⊢ (𝐹 ∈ Field → 𝐹 ∈ Ring) |
16 | ssidd 4005 | . . . . . . 7 ⊢ (𝐹 ∈ Field → (Base‘𝐹) ⊆ (Base‘𝐹)) | |
17 | 11, 3 | sraring 21038 | . . . . . . 7 ⊢ ((𝐹 ∈ Ring ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ Ring) |
18 | 15, 16, 17 | syl2anc 583 | . . . . . 6 ⊢ (𝐹 ∈ Field → 𝑉 ∈ Ring) |
19 | eqid 2731 | . . . . . . 7 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
20 | eqid 2731 | . . . . . . 7 ⊢ (1r‘𝑉) = (1r‘𝑉) | |
21 | 19, 20 | ringidcl 20161 | . . . . . 6 ⊢ (𝑉 ∈ Ring → (1r‘𝑉) ∈ (Base‘𝑉)) |
22 | 18, 21 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Field → (1r‘𝑉) ∈ (Base‘𝑉)) |
23 | 11, 3 | sradrng 33126 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (Base‘𝐹) ⊆ (Base‘𝐹)) → 𝑉 ∈ DivRing) |
24 | 2, 16, 23 | syl2anc 583 | . . . . . 6 ⊢ (𝐹 ∈ Field → 𝑉 ∈ DivRing) |
25 | eqid 2731 | . . . . . . 7 ⊢ (0g‘𝑉) = (0g‘𝑉) | |
26 | 25, 20 | drngunz 20602 | . . . . . 6 ⊢ (𝑉 ∈ DivRing → (1r‘𝑉) ≠ (0g‘𝑉)) |
27 | 24, 26 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ Field → (1r‘𝑉) ≠ (0g‘𝑉)) |
28 | 19, 25 | lindssn 32936 | . . . . 5 ⊢ ((𝑉 ∈ LVec ∧ (1r‘𝑉) ∈ (Base‘𝑉) ∧ (1r‘𝑉) ≠ (0g‘𝑉)) → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
29 | 14, 22, 27, 28 | syl3anc 1370 | . . . 4 ⊢ (𝐹 ∈ Field → {(1r‘𝑉)} ∈ (LIndS‘𝑉)) |
30 | rspval 21066 | . . . . . . . . 9 ⊢ (RSpan‘𝐹) = (LSpan‘(ringLMod‘𝐹)) | |
31 | 9 | fveq2i 6894 | . . . . . . . . 9 ⊢ (LSpan‘𝑉) = (LSpan‘(ringLMod‘𝐹)) |
32 | 30, 31 | eqtr4i 2762 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (LSpan‘𝑉) |
33 | 32 | fveq1i 6892 | . . . . . . 7 ⊢ ((RSpan‘𝐹)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝐹)}) |
34 | eqid 2731 | . . . . . . . 8 ⊢ (RSpan‘𝐹) = (RSpan‘𝐹) | |
35 | eqid 2731 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
36 | 34, 3, 35 | rsp1 21092 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → ((RSpan‘𝐹)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
37 | 33, 36 | eqtr3id 2785 | . . . . . 6 ⊢ (𝐹 ∈ Ring → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
38 | 2, 6, 37 | 3syl 18 | . . . . 5 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = (Base‘𝐹)) |
39 | 11 | a1i 11 | . . . . . . . 8 ⊢ (𝐹 ∈ Field → 𝑉 = ((subringAlg ‘𝐹)‘(Base‘𝐹))) |
40 | eqidd 2732 | . . . . . . . 8 ⊢ (𝐹 ∈ Field → (1r‘𝐹) = (1r‘𝐹)) | |
41 | 39, 40, 16 | sra1r 33125 | . . . . . . 7 ⊢ (𝐹 ∈ Field → (1r‘𝐹) = (1r‘𝑉)) |
42 | 41 | sneqd 4640 | . . . . . 6 ⊢ (𝐹 ∈ Field → {(1r‘𝐹)} = {(1r‘𝑉)}) |
43 | 42 | fveq2d 6895 | . . . . 5 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝐹)}) = ((LSpan‘𝑉)‘{(1r‘𝑉)})) |
44 | 39, 16 | srabase 21022 | . . . . 5 ⊢ (𝐹 ∈ Field → (Base‘𝐹) = (Base‘𝑉)) |
45 | 38, 43, 44 | 3eqtr3d 2779 | . . . 4 ⊢ (𝐹 ∈ Field → ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉)) |
46 | eqid 2731 | . . . . 5 ⊢ (LBasis‘𝑉) = (LBasis‘𝑉) | |
47 | eqid 2731 | . . . . 5 ⊢ (LSpan‘𝑉) = (LSpan‘𝑉) | |
48 | 19, 46, 47 | islbs4 21698 | . . . 4 ⊢ ({(1r‘𝑉)} ∈ (LBasis‘𝑉) ↔ ({(1r‘𝑉)} ∈ (LIndS‘𝑉) ∧ ((LSpan‘𝑉)‘{(1r‘𝑉)}) = (Base‘𝑉))) |
49 | 29, 45, 48 | sylanbrc 582 | . . 3 ⊢ (𝐹 ∈ Field → {(1r‘𝑉)} ∈ (LBasis‘𝑉)) |
50 | 46 | dimval 33141 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ {(1r‘𝑉)} ∈ (LBasis‘𝑉)) → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
51 | 14, 49, 50 | syl2anc 583 | . 2 ⊢ (𝐹 ∈ Field → (dim‘𝑉) = (♯‘{(1r‘𝑉)})) |
52 | fvex 6904 | . . 3 ⊢ (1r‘𝑉) ∈ V | |
53 | hashsng 14336 | . . 3 ⊢ ((1r‘𝑉) ∈ V → (♯‘{(1r‘𝑉)}) = 1) | |
54 | 52, 53 | ax-mp 5 | . 2 ⊢ (♯‘{(1r‘𝑉)}) = 1 |
55 | 51, 54 | eqtrdi 2787 | 1 ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ⊆ wss 3948 {csn 4628 ‘cfv 6543 (class class class)co 7412 1c1 11117 ♯chash 14297 Basecbs 17151 ↾s cress 17180 0gc0g 17392 1rcur 20082 Ringcrg 20134 CRingccrg 20135 SubRingcsubrg 20465 DivRingcdr 20583 Fieldcfield 20584 LSpanclspn 20814 LBasisclbs 20918 LVecclvec 20946 subringAlg csra 21015 ringLModcrglmod 21016 RSpancrsp 21062 LIndSclinds 21671 dimcldim 33139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-reg 9593 ax-inf2 9642 ax-ac2 10464 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-oi 9511 df-r1 9765 df-rank 9766 df-card 9940 df-acn 9943 df-ac 10117 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-xnn0 12552 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ocomp 17225 df-0g 17394 df-mre 17537 df-mrc 17538 df-mri 17539 df-acs 17540 df-proset 18258 df-drs 18259 df-poset 18276 df-ipo 18491 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-subrg 20467 df-drng 20585 df-field 20586 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lbs 20919 df-lvec 20947 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-rsp 21064 df-lindf 21672 df-linds 21673 df-dim 33140 |
This theorem is referenced by: (None) |
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