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Mirrors > Home > MPE Home > Th. List > cnmsubglem | Structured version Visualization version GIF version |
Description: Lemma for rpmsubg 20291 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 4626 | . . . 4 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
4 | 1, 2, 3 | sylanbrc 583 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (ℂ ∖ {0})) |
5 | 4 | ssriv 3893 | . 2 ⊢ 𝐴 ⊆ (ℂ ∖ {0}) |
6 | cnmsubglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
7 | 6 | ne0ii 4223 | . 2 ⊢ 𝐴 ≠ ∅ |
8 | cnmsubglem.3 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
9 | 8 | ralrimiva 3149 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) |
10 | cnfldinv 20258 | . . . . . 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 2883 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
14 | 9, 13 | jca 512 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴)) |
15 | 14 | rgen 3115 | . 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 20288 | . . 3 ⊢ 𝑀 ∈ Abel |
18 | ablgrp 18638 | . . 3 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
19 | difss 4029 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
20 | eqid 2795 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 20231 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 18935 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 16, 22 | ressbas2 16384 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑀)) |
24 | 19, 23 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑀) |
25 | cnex 10464 | . . . . 5 ⊢ ℂ ∈ V | |
26 | difexg 5122 | . . . . 5 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V) | |
27 | cnfldmul 20233 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
28 | 20, 27 | mgpplusg 18933 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 16, 28 | ressplusg 16441 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑀)) |
30 | 25, 26, 29 | mp2b 10 | . . . 4 ⊢ · = (+g‘𝑀) |
31 | cnfld0 20251 | . . . . . 6 ⊢ 0 = (0g‘ℂfld) | |
32 | cndrng 20256 | . . . . . 6 ⊢ ℂfld ∈ DivRing | |
33 | 21, 31, 32 | drngui 19198 | . . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
34 | eqid 2795 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
35 | 33, 16, 34 | invrfval 19113 | . . . 4 ⊢ (invr‘ℂfld) = (invg‘𝑀) |
36 | 24, 30, 35 | issubg2 18048 | . . 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 1334 | 1 ⊢ 𝐴 ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 Vcvv 3437 ∖ cdif 3856 ⊆ wss 3859 ∅c0 4211 {csn 4472 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 0cc0 10383 1c1 10384 · cmul 10388 / cdiv 11145 Basecbs 16312 ↾s cress 16313 +gcplusg 16394 Grpcgrp 17861 SubGrpcsubg 18027 Abelcabl 18634 mulGrpcmgp 18929 invrcinvr 19111 ℂfldccnfld 20227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 df-minusg 17865 df-subg 18030 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-cnfld 20228 |
This theorem is referenced by: rpmsubg 20291 cnmsgnsubg 20403 |
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