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| Mirrors > Home > MPE Home > Th. List > cnsubdrglem | Structured version Visualization version GIF version | ||
| Description: Lemma for resubdrg 21626 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 21434 | . 2 ⊢ 𝐴 ∈ (SubRing‘ℂfld) |
| 7 | cndrng 21411 | . . . 4 ⊢ ℂfld ∈ DivRing | |
| 8 | eqid 2737 | . . . . 5 ⊢ (ℂfld ↾s 𝐴) = (ℂfld ↾s 𝐴) | |
| 9 | cnfld0 21405 | . . . . 5 ⊢ 0 = (0g‘ℂfld) | |
| 10 | eqid 2737 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 11 | 8, 9, 10 | issubdrg 20781 | . . . 4 ⊢ ((ℂfld ∈ DivRing ∧ 𝐴 ∈ (SubRing‘ℂfld)) → ((ℂfld ↾s 𝐴) ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ {0})((invr‘ℂfld)‘𝑥) ∈ 𝐴)) |
| 12 | 7, 6, 11 | mp2an 692 | . . 3 ⊢ ((ℂfld ↾s 𝐴) ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ {0})((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
| 13 | cnring 21403 | . . . . 5 ⊢ ℂfld ∈ Ring | |
| 14 | 1 | ssriv 3987 | . . . . . . 7 ⊢ 𝐴 ⊆ ℂ |
| 15 | ssdif 4144 | . . . . . . 7 ⊢ (𝐴 ⊆ ℂ → (𝐴 ∖ {0}) ⊆ (ℂ ∖ {0})) | |
| 16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 ∖ {0}) ⊆ (ℂ ∖ {0}) |
| 17 | 16 | sseli 3979 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → 𝑥 ∈ (ℂ ∖ {0})) |
| 18 | cnfldbas 21368 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 19 | 18, 9, 7 | drngui 20735 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
| 20 | cnflddiv 21413 | . . . . . 6 ⊢ / = (/r‘ℂfld) | |
| 21 | cnfld1 21406 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
| 22 | 18, 19, 20, 21, 10 | ringinvdv 20414 | . . . . 5 ⊢ ((ℂfld ∈ Ring ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
| 23 | 13, 17, 22 | sylancr 587 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
| 24 | eldifsn 4786 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0)) | |
| 25 | cnsubrglem.6 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐴) | |
| 26 | 24, 25 | sylbi 217 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → (1 / 𝑥) ∈ 𝐴) |
| 27 | 23, 26 | eqeltrd 2841 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {0}) → ((invr‘ℂfld)‘𝑥) ∈ 𝐴) |
| 28 | 12, 27 | mprgbir 3068 | . 2 ⊢ (ℂfld ↾s 𝐴) ∈ DivRing |
| 29 | 6, 28 | pm3.2i 470 | 1 ⊢ (𝐴 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐴) ∈ DivRing) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 -cneg 11493 / cdiv 11920 ↾s cress 17274 Ringcrg 20230 invrcinvr 20387 SubRingcsubrg 20569 DivRingcdr 20729 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-cnfld 21365 |
| This theorem is referenced by: qsubdrg 21437 resubdrg 21626 |
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