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Mirrors > Home > MPE Home > Th. List > cnmsubglem | Structured version Visualization version GIF version |
Description: Lemma for rpmsubg 21421 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
cnmgpabl.m | ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
cnmsubglem.1 | ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) |
cnmsubglem.2 | ⊢ (𝑥 ∈ 𝐴 → 𝑥 ≠ 0) |
cnmsubglem.3 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) |
cnmsubglem.4 | ⊢ 1 ∈ 𝐴 |
cnmsubglem.5 | ⊢ (𝑥 ∈ 𝐴 → (1 / 𝑥) ∈ 𝐴) |
Ref | Expression |
---|---|
cnmsubglem | ⊢ 𝐴 ∈ (SubGrp‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmsubglem.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) | |
2 | cnmsubglem.2 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ≠ 0) | |
3 | eldifsn 4785 | . . . 4 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
4 | 1, 2, 3 | sylanbrc 581 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (ℂ ∖ {0})) |
5 | 4 | ssriv 3982 | . 2 ⊢ 𝐴 ⊆ (ℂ ∖ {0}) |
6 | cnmsubglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
7 | 6 | ne0ii 4337 | . 2 ⊢ 𝐴 ≠ ∅ |
8 | cnmsubglem.3 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
9 | 8 | ralrimiva 3136 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) |
10 | cnfldinv 21387 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) | |
11 | 1, 2, 10 | syl2anc 582 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
12 | cnmsubglem.5 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (1 / 𝑥) ∈ 𝐴) | |
13 | 11, 12 | eqeltrd 2826 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
14 | 9, 13 | jca 510 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴)) |
15 | 14 | rgen 3053 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
16 | cnmgpabl.m | . . . 4 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
17 | 16 | cnmgpabl 21418 | . . 3 ⊢ 𝑀 ∈ Abel |
18 | ablgrp 19776 | . . 3 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
19 | difss 4128 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
20 | eqid 2726 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 21340 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 20116 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 16, 22 | ressbas2 17243 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑀)) |
24 | 19, 23 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑀) |
25 | cnex 11227 | . . . . 5 ⊢ ℂ ∈ V | |
26 | difexg 5324 | . . . . 5 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V) | |
27 | cnfldmul 21344 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
28 | 20, 27 | mgpplusg 20114 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 16, 28 | ressplusg 17296 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑀)) |
30 | 25, 26, 29 | mp2b 10 | . . . 4 ⊢ · = (+g‘𝑀) |
31 | cnfld0 21377 | . . . . . 6 ⊢ 0 = (0g‘ℂfld) | |
32 | cndrng 21383 | . . . . . 6 ⊢ ℂfld ∈ DivRing | |
33 | 21, 31, 32 | drngui 20706 | . . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
34 | eqid 2726 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
35 | 33, 16, 34 | invrfval 20364 | . . . 4 ⊢ (invr‘ℂfld) = (invg‘𝑀) |
36 | 24, 30, 35 | issubg2 19128 | . . 3 ⊢ (𝑀 ∈ Grp → (𝐴 ∈ (SubGrp‘𝑀) ↔ (𝐴 ⊆ (ℂ ∖ {0}) ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴)))) |
37 | 17, 18, 36 | mp2b 10 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑀) ↔ (𝐴 ⊆ (ℂ ∖ {0}) ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴))) |
38 | 5, 7, 15, 37 | mpbir3an 1338 | 1 ⊢ 𝐴 ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 Vcvv 3462 ∖ cdif 3943 ⊆ wss 3946 ∅c0 4322 {csn 4623 ‘cfv 6543 (class class class)co 7413 ℂcc 11144 0cc0 11146 1c1 11147 · cmul 11151 / cdiv 11909 Basecbs 17205 ↾s cress 17234 +gcplusg 17258 Grpcgrp 18920 SubGrpcsubg 19107 Abelcabl 19772 mulGrpcmgp 20110 invrcinvr 20362 ℂfldccnfld 21336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 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 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12516 df-z 12602 df-dec 12721 df-uz 12866 df-fz 13530 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-ress 17235 df-plusg 17271 df-mulr 17272 df-starv 17273 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-0g 17448 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-grp 18923 df-minusg 18924 df-subg 19110 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20309 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20702 df-cnfld 21337 |
This theorem is referenced by: rpmsubg 21421 cnmsgnsubg 21566 |
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