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Mirrors > Home > MPE Home > Th. List > cnsubdrglem | Structured version Visualization version GIF version |
Description: Lemma for resubdrg 20746 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) |
cnsubrglem.6 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐴) |
Ref | Expression |
---|---|
cnsubdrglem | ⊢ (𝐴 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐴) ∈ DivRing) |
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 | cnsubrglem.5 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
6 | 1, 2, 3, 4, 5 | cnsubrglem 20589 | . 2 ⊢ 𝐴 ∈ (SubRing‘ℂfld) |
7 | cndrng 20568 | . . . 4 ⊢ ℂfld ∈ DivRing | |
8 | eqid 2821 | . . . . 5 ⊢ (ℂfld ↾s 𝐴) = (ℂfld ↾s 𝐴) | |
9 | cnfld0 20563 | . . . . 5 ⊢ 0 = (0g‘ℂfld) | |
10 | eqid 2821 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
11 | 8, 9, 10 | issubdrg 19554 | . . . 4 ⊢ ((ℂfld ∈ DivRing ∧ 𝐴 ∈ (SubRing‘ℂfld)) → ((ℂfld ↾s 𝐴) ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ {0})((invr‘ℂfld)‘𝑥) ∈ 𝐴)) |
12 | 7, 6, 11 | mp2an 690 | . . 3 ⊢ ((ℂfld ↾s 𝐴) ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ {0})((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
13 | cnring 20561 | . . . . 5 ⊢ ℂfld ∈ Ring | |
14 | 1 | ssriv 3971 | . . . . . . 7 ⊢ 𝐴 ⊆ ℂ |
15 | ssdif 4116 | . . . . . . 7 ⊢ (𝐴 ⊆ ℂ → (𝐴 ∖ {0}) ⊆ (ℂ ∖ {0})) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 ∖ {0}) ⊆ (ℂ ∖ {0}) |
17 | 16 | sseli 3963 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → 𝑥 ∈ (ℂ ∖ {0})) |
18 | cnfldbas 20543 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
19 | 18, 9, 7 | drngui 19502 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
20 | cnflddiv 20569 | . . . . . 6 ⊢ / = (/r‘ℂfld) | |
21 | cnfld1 20564 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
22 | 18, 19, 20, 21, 10 | ringinvdv 19438 | . . . . 5 ⊢ ((ℂfld ∈ Ring ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
23 | 13, 17, 22 | sylancr 589 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
24 | eldifsn 4713 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0)) | |
25 | cnsubrglem.6 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐴) | |
26 | 24, 25 | sylbi 219 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → (1 / 𝑥) ∈ 𝐴) |
27 | 23, 26 | eqeltrd 2913 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
28 | 12, 27 | mprgbir 3153 | . 2 ⊢ (ℂfld ↾s 𝐴) ∈ DivRing |
29 | 6, 28 | pm3.2i 473 | 1 ⊢ (𝐴 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐴) ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∖ cdif 3933 ⊆ wss 3936 {csn 4561 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 -cneg 10865 / cdiv 11291 ↾s cress 16478 Ringcrg 19291 invrcinvr 19415 DivRingcdr 19496 SubRingcsubrg 19525 ℂfldccnfld 20539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-subrg 19527 df-cnfld 20540 |
This theorem is referenced by: qsubdrg 20591 resubdrg 20746 |
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