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Mirrors > Home > MPE Home > Th. List > cnsubglem | Structured version Visualization version GIF version |
Description: Lemma for resubdrg 21035 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 3952 | . 2 ⊢ 𝐴 ⊆ ℂ |
3 | cnsubglem.4 | . . 3 ⊢ 𝐵 ∈ 𝐴 | |
4 | 3 | ne0ii 4301 | . 2 ⊢ 𝐴 ≠ ∅ |
5 | cnsubglem.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) | |
6 | 5 | ralrimiva 3140 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴) |
7 | cnfldneg 20846 | . . . . . 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 2834 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invg‘ℂfld)‘𝑥) ∈ 𝐴) |
11 | 6, 10 | jca 513 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴)) |
12 | 11 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴) |
13 | cnring 20842 | . . 3 ⊢ ℂfld ∈ Ring | |
14 | ringgrp 19977 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
15 | cnfldbas 20823 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
16 | cnfldadd 20824 | . . . 4 ⊢ + = (+g‘ℂfld) | |
17 | eqid 2733 | . . . 4 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
18 | 15, 16, 17 | issubg2 18951 | . . 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 1342 | 1 ⊢ 𝐴 ∈ (SubGrp‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ⊆ wss 3914 ∅c0 4286 ‘cfv 6500 (class class class)co 7361 ℂcc 11057 + caddc 11062 -cneg 11394 Grpcgrp 18756 invgcminusg 18757 SubGrpcsubg 18930 Ringcrg 19972 ℂfldccnfld 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-subg 18933 df-cmn 19572 df-mgp 19905 df-ring 19974 df-cring 19975 df-cnfld 20820 |
This theorem is referenced by: cnsubrglem 20870 zringmulg 20900 remulg 21034 |
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