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Mirrors > Home > MPE Home > Th. List > cnsubglem | Structured version Visualization version GIF version |
Description: Lemma for resubdrg 20959 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
cnsubglem.1 | ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) |
cnsubglem.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) |
cnsubglem.3 | ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) |
cnsubglem.4 | ⊢ 𝐵 ∈ 𝐴 |
Ref | Expression |
---|---|
cnsubglem | ⊢ 𝐴 ∈ (SubGrp‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsubglem.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) | |
2 | 1 | ssriv 3946 | . 2 ⊢ 𝐴 ⊆ ℂ |
3 | cnsubglem.4 | . . 3 ⊢ 𝐵 ∈ 𝐴 | |
4 | 3 | ne0ii 4295 | . 2 ⊢ 𝐴 ≠ ∅ |
5 | cnsubglem.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) | |
6 | 5 | ralrimiva 3141 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴) |
7 | cnfldneg 20770 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((invg‘ℂfld)‘𝑥) = -𝑥) |
9 | cnsubglem.3 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) | |
10 | 8, 9 | eqeltrd 2838 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invg‘ℂfld)‘𝑥) ∈ 𝐴) |
11 | 6, 10 | jca 512 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴)) |
12 | 11 | rgen 3064 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴) |
13 | cnring 20766 | . . 3 ⊢ ℂfld ∈ Ring | |
14 | ringgrp 19917 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
15 | cnfldbas 20747 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
16 | cnfldadd 20748 | . . . 4 ⊢ + = (+g‘ℂfld) | |
17 | eqid 2736 | . . . 4 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
18 | 15, 16, 17 | issubg2 18896 | . . 3 ⊢ (ℂfld ∈ Grp → (𝐴 ∈ (SubGrp‘ℂfld) ↔ (𝐴 ⊆ ℂ ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴)))) |
19 | 13, 14, 18 | mp2b 10 | . 2 ⊢ (𝐴 ∈ (SubGrp‘ℂfld) ↔ (𝐴 ⊆ ℂ ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴))) |
20 | 2, 4, 12, 19 | mpbir3an 1341 | 1 ⊢ 𝐴 ∈ (SubGrp‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ⊆ wss 3908 ∅c0 4280 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 + caddc 11012 -cneg 11344 Grpcgrp 18702 invgcminusg 18703 SubGrpcsubg 18875 Ringcrg 19912 ℂfldccnfld 20743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-starv 17102 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-0g 17277 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-grp 18705 df-minusg 18706 df-subg 18878 df-cmn 19517 df-mgp 19850 df-ring 19914 df-cring 19915 df-cnfld 20744 |
This theorem is referenced by: cnsubrglem 20794 zringmulg 20824 remulg 20958 |
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