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Mirrors > Home > MPE Home > Th. List > cnmsubglem | Structured version Visualization version GIF version |
Description: Lemma for rpmsubg 21466 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 4790 | . . . 4 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
4 | 1, 2, 3 | sylanbrc 583 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (ℂ ∖ {0})) |
5 | 4 | ssriv 3998 | . 2 ⊢ 𝐴 ⊆ (ℂ ∖ {0}) |
6 | cnmsubglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
7 | 6 | ne0ii 4349 | . 2 ⊢ 𝐴 ≠ ∅ |
8 | cnmsubglem.3 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
9 | 8 | ralrimiva 3143 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) |
10 | cnfldinv 21432 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) | |
11 | 1, 2, 10 | syl2anc 584 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
12 | cnmsubglem.5 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (1 / 𝑥) ∈ 𝐴) | |
13 | 11, 12 | eqeltrd 2838 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
14 | 9, 13 | jca 511 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴)) |
15 | 14 | rgen 3060 | . 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 21463 | . . 3 ⊢ 𝑀 ∈ Abel |
18 | ablgrp 19817 | . . 3 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
19 | difss 4145 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
20 | eqid 2734 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 21385 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 20157 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 16, 22 | ressbas2 17282 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑀)) |
24 | 19, 23 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑀) |
25 | cnex 11233 | . . . . 5 ⊢ ℂ ∈ V | |
26 | difexg 5334 | . . . . 5 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V) | |
27 | cnfldmul 21389 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
28 | 20, 27 | mgpplusg 20155 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 16, 28 | ressplusg 17335 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑀)) |
30 | 25, 26, 29 | mp2b 10 | . . . 4 ⊢ · = (+g‘𝑀) |
31 | cnfld0 21422 | . . . . . 6 ⊢ 0 = (0g‘ℂfld) | |
32 | cndrng 21428 | . . . . . 6 ⊢ ℂfld ∈ DivRing | |
33 | 21, 31, 32 | drngui 20751 | . . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
34 | eqid 2734 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
35 | 33, 16, 34 | invrfval 20405 | . . . 4 ⊢ (invr‘ℂfld) = (invg‘𝑀) |
36 | 24, 30, 35 | issubg2 19171 | . . 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 1340 | 1 ⊢ 𝐴 ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 ∅c0 4338 {csn 4630 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 · cmul 11157 / cdiv 11917 Basecbs 17244 ↾s cress 17273 +gcplusg 17297 Grpcgrp 18963 SubGrpcsubg 19150 Abelcabl 19813 mulGrpcmgp 20151 invrcinvr 20403 ℂfldccnfld 21381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 ax-mulf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-drng 20747 df-cnfld 21382 |
This theorem is referenced by: rpmsubg 21466 cnmsgnsubg 21612 |
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