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| Mirrors > Home > MPE Home > Th. List > cnmsubglem | Structured version Visualization version GIF version | ||
| Description: Lemma for rpmsubg 21449 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 4786 | . . . 4 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
| 4 | 1, 2, 3 | sylanbrc 583 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (ℂ ∖ {0})) |
| 5 | 4 | ssriv 3987 | . 2 ⊢ 𝐴 ⊆ (ℂ ∖ {0}) |
| 6 | cnmsubglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
| 7 | 6 | ne0ii 4344 | . 2 ⊢ 𝐴 ≠ ∅ |
| 8 | cnmsubglem.3 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
| 9 | 8 | ralrimiva 3146 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) |
| 10 | cnfldinv 21415 | . . . . . 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 2841 | . . . 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 3063 | . 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 21446 | . . 3 ⊢ 𝑀 ∈ Abel |
| 18 | ablgrp 19803 | . . 3 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
| 19 | difss 4136 | . . . . 5 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 21 | cnfldbas 21368 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 22 | 20, 21 | mgpbas 20142 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 23 | 16, 22 | ressbas2 17283 | . . . . 5 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (ℂ ∖ {0}) = (Base‘𝑀)) |
| 24 | 19, 23 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) = (Base‘𝑀) |
| 25 | cnex 11236 | . . . . 5 ⊢ ℂ ∈ V | |
| 26 | difexg 5329 | . . . . 5 ⊢ (ℂ ∈ V → (ℂ ∖ {0}) ∈ V) | |
| 27 | cnfldmul 21372 | . . . . . . 7 ⊢ · = (.r‘ℂfld) | |
| 28 | 20, 27 | mgpplusg 20141 | . . . . . 6 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 29 | 16, 28 | ressplusg 17334 | . . . . 5 ⊢ ((ℂ ∖ {0}) ∈ V → · = (+g‘𝑀)) |
| 30 | 25, 26, 29 | mp2b 10 | . . . 4 ⊢ · = (+g‘𝑀) |
| 31 | cnfld0 21405 | . . . . . 6 ⊢ 0 = (0g‘ℂfld) | |
| 32 | cndrng 21411 | . . . . . 6 ⊢ ℂfld ∈ DivRing | |
| 33 | 21, 31, 32 | drngui 20735 | . . . . 5 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
| 34 | eqid 2737 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 35 | 33, 16, 34 | invrfval 20389 | . . . 4 ⊢ (invr‘ℂfld) = (invg‘𝑀) |
| 36 | 24, 30, 35 | issubg2 19159 | . . 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 1342 | 1 ⊢ 𝐴 ∈ (SubGrp‘𝑀) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 {csn 4626 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 Grpcgrp 18951 SubGrpcsubg 19138 Abelcabl 19799 mulGrpcmgp 20137 invrcinvr 20387 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-cnfld 21365 |
| This theorem is referenced by: rpmsubg 21449 cnmsgnsubg 21595 |
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