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Mirrors > Home > ILE Home > Th. List > cnsubglem | GIF version |
Description: Lemma for cnsubrglem 13343 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 3159 | . 2 ⊢ 𝐴 ⊆ ℂ |
3 | cnsubglem.4 | . . 3 ⊢ 𝐵 ∈ 𝐴 | |
4 | elex2 2753 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ∃𝑤 𝑤 ∈ 𝐴 |
6 | cnsubglem.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) | |
7 | 6 | ralrimiva 2550 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴) |
8 | cnfldneg 13336 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
9 | 1, 8 | syl 14 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((invg‘ℂfld)‘𝑥) = -𝑥) |
10 | cnsubglem.3 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) | |
11 | 9, 10 | eqeltrd 2254 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((invg‘ℂfld)‘𝑥) ∈ 𝐴) |
12 | 7, 11 | jca 306 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴)) |
13 | 12 | rgen 2530 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴) |
14 | cnring 13333 | . . 3 ⊢ ℂfld ∈ Ring | |
15 | ringgrp 13115 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
16 | cnfldbas 13328 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
17 | cnfldadd 13329 | . . . 4 ⊢ + = (+g‘ℂfld) | |
18 | eqid 2177 | . . . 4 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
19 | 16, 17, 18 | issubg2m 12980 | . . 3 ⊢ (ℂfld ∈ Grp → (𝐴 ∈ (SubGrp‘ℂfld) ↔ (𝐴 ⊆ ℂ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴)))) |
20 | 14, 15, 19 | mp2b 8 | . 2 ⊢ (𝐴 ∈ (SubGrp‘ℂfld) ↔ (𝐴 ⊆ ℂ ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 + 𝑦) ∈ 𝐴 ∧ ((invg‘ℂfld)‘𝑥) ∈ 𝐴))) |
21 | 2, 5, 13, 20 | mpbir3an 1179 | 1 ⊢ 𝐴 ∈ (SubGrp‘ℂfld) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 ‘cfv 5215 (class class class)co 5872 ℂcc 7806 + caddc 7811 -cneg 8125 Grpcgrp 12809 invgcminusg 12810 SubGrpcsubg 12958 Ringcrg 13110 ℂfldccnfld 13324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-addf 7930 ax-mulf 7931 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-tp 3600 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-5 8977 df-6 8978 df-7 8979 df-8 8980 df-9 8981 df-n0 9173 df-z 9250 df-dec 9381 df-uz 9525 df-fz 10005 df-cj 10844 df-struct 12456 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-mulr 12542 df-starv 12543 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-subg 12961 df-cmn 13021 df-mgp 13062 df-ring 13112 df-cring 13113 df-icnfld 13325 |
This theorem is referenced by: cnsubrglem 13343 zringmulg 13357 |
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