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Mirrors > Home > MPE Home > Th. List > cnmsubglem | Structured version Visualization version GIF version |
Description: Lemma for rpmsubg 19629 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 4260 | . . . 4 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
4 | 1, 2, 3 | sylanbrc 695 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (ℂ ∖ {0})) |
5 | 4 | ssriv 3572 | . 2 ⊢ 𝐴 ⊆ (ℂ ∖ {0}) |
6 | cnmsubglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
7 | 6 | ne0ii 3882 | . 2 ⊢ 𝐴 ≠ ∅ |
8 | cnmsubglem.3 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
9 | 8 | ralrimiva 2949 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) |
10 | cnfldinv 19596 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) | |
11 | 1, 2, 10 | syl2anc 691 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
12 | cnmsubglem.5 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (1 / 𝑥) ∈ 𝐴) | |
13 | 11, 12 | eqeltrd 2688 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
14 | 9, 13 | jca 553 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 ∧ ((invr‘ℂfld)‘𝑥) ∈ 𝐴)) |
15 | 14 | rgen 2906 | . 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 19626 | . . 3 ⊢ 𝑀 ∈ Abel |
18 | ablgrp 18021 | . . 3 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
19 | difss 3699 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
20 | eqid 2610 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
21 | cnfldbas 19571 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
22 | 20, 21 | mgpbas 18318 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
23 | 16, 22 | ressbas2 15758 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑀)) |
24 | 19, 23 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑀) |
25 | cnex 9896 | . . . . 5 ⊢ ℂ ∈ V | |
26 | difexg 4735 | . . . . 5 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V) | |
27 | cnfldmul 19573 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
28 | 20, 27 | mgpplusg 18316 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
29 | 16, 28 | ressplusg 15818 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑀)) |
30 | 25, 26, 29 | mp2b 10 | . . . 4 ⊢ · = (+g‘𝑀) |
31 | cnfld0 19589 | . . . . . 6 ⊢ 0 = (0g‘ℂfld) | |
32 | cndrng 19594 | . . . . . 6 ⊢ ℂfld ∈ DivRing | |
33 | 21, 31, 32 | drngui 18576 | . . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
34 | eqid 2610 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
35 | 33, 16, 34 | invrfval 18496 | . . . 4 ⊢ (invr‘ℂfld) = (invg‘𝑀) |
36 | 24, 30, 35 | issubg2 17432 | . . 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 1237 | 1 ⊢ 𝐴 ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 Grpcgrp 17245 SubGrpcsubg 17411 Abelcabl 18017 mulGrpcmgp 18312 invrcinvr 18494 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-cnfld 19568 |
This theorem is referenced by: rpmsubg 19629 cnmsgnsubg 19742 |
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