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Mirrors > Home > ILE Home > Th. List > cnsubrglem | GIF version |
Description: Lemma for zsubrg 13757 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) |
cnsubrglem.4 | ⊢ 1 ∈ 𝐴 |
cnsubrglem.5 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) |
Ref | Expression |
---|---|
cnsubrglem | ⊢ 𝐴 ∈ (SubRing‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsubglem.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) | |
2 | cnsubglem.2 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) | |
3 | cnsubglem.3 | . . 3 ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) | |
4 | cnsubrglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
5 | 1, 2, 3, 4 | cnsubglem 13755 | . 2 ⊢ 𝐴 ∈ (SubGrp‘ℂfld) |
6 | cnsubrglem.5 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
7 | 6 | rgen2 2573 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 |
8 | cnring 13746 | . . 3 ⊢ ℂfld ∈ Ring | |
9 | cnfldbas 13741 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
10 | cnfld1 13748 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
11 | cnfldmul 13743 | . . . 4 ⊢ · = (.r‘ℂfld) | |
12 | 9, 10, 11 | issubrg2 13461 | . . 3 ⊢ (ℂfld ∈ Ring → (𝐴 ∈ (SubRing‘ℂfld) ↔ (𝐴 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))) |
13 | 8, 12 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ (SubRing‘ℂfld) ↔ (𝐴 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) |
14 | 5, 4, 7, 13 | mpbir3an 1180 | 1 ⊢ 𝐴 ∈ (SubRing‘ℂfld) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 979 ∈ wcel 2158 ∀wral 2465 ‘cfv 5228 (class class class)co 5888 ℂcc 7823 1c1 7826 + caddc 7828 · cmul 7830 -cneg 8143 SubGrpcsubg 13059 Ringcrg 13248 SubRingcsubrg 13437 ℂfldccnfld 13737 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-addf 7947 ax-mulf 7948 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-tp 3612 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-7 8997 df-8 8998 df-9 8999 df-n0 9191 df-z 9268 df-dec 9399 df-uz 9543 df-fz 10023 df-cj 10865 df-struct 12478 df-ndx 12479 df-slot 12480 df-base 12482 df-sets 12483 df-iress 12484 df-plusg 12564 df-mulr 12565 df-starv 12566 df-0g 12725 df-mgm 12794 df-sgrp 12827 df-mnd 12840 df-grp 12902 df-minusg 12903 df-subg 13062 df-cmn 13123 df-mgp 13173 df-ur 13212 df-ring 13250 df-cring 13251 df-subrg 13439 df-icnfld 13738 |
This theorem is referenced by: zsubrg 13757 gzsubrg 13758 |
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