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Mirrors > Home > MPE Home > Th. List > cphreccllem | Structured version Visualization version GIF version |
Description: Lemma for cphreccl 23785. (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
cphsubrglem.k | ⊢ 𝐾 = (Base‘𝐹) |
cphsubrglem.1 | ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐴)) |
cphsubrglem.2 | ⊢ (𝜑 → 𝐹 ∈ DivRing) |
Ref | Expression |
---|---|
cphreccllem | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphsubrglem.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
2 | cphsubrglem.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐴)) | |
3 | cphsubrglem.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ DivRing) | |
4 | 1, 2, 3 | cphsubrglem 23781 | . . . . . . 7 ⊢ (𝜑 → (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
5 | 4 | simp3d 1140 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubRing‘ℂfld)) |
6 | 5 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐾 ∈ (SubRing‘ℂfld)) |
7 | cnfldbas 20549 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
8 | 7 | subrgss 19536 | . . . . 5 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ⊆ ℂ) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐾 ⊆ ℂ) |
10 | simp2 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ 𝐾) | |
11 | 9, 10 | sseldd 3968 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ ℂ) |
12 | simp3 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ≠ 0) | |
13 | cnfldinv 20576 | . . 3 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋)) | |
14 | 11, 12, 13 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋)) |
15 | eqid 2821 | . . . . . . . . . 10 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
16 | cnfld0 20569 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
17 | 15, 16 | subrg0 19542 | . . . . . . . . 9 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂfld ↾s 𝐾))) |
18 | 6, 17 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 0 = (0g‘(ℂfld ↾s 𝐾))) |
19 | 4 | simp1d 1138 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) |
20 | 19 | 3ad2ant1 1129 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐹 = (ℂfld ↾s 𝐾)) |
21 | 20 | fveq2d 6674 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (0g‘𝐹) = (0g‘(ℂfld ↾s 𝐾))) |
22 | 18, 21 | eqtr4d 2859 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 0 = (0g‘𝐹)) |
23 | 12, 22 | neeqtrd 3085 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ≠ (0g‘𝐹)) |
24 | eldifsn 4719 | . . . . . 6 ⊢ (𝑋 ∈ (𝐾 ∖ {(0g‘𝐹)}) ↔ (𝑋 ∈ 𝐾 ∧ 𝑋 ≠ (0g‘𝐹))) | |
25 | 10, 23, 24 | sylanbrc 585 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ (𝐾 ∖ {(0g‘𝐹)})) |
26 | 3 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐹 ∈ DivRing) |
27 | eqid 2821 | . . . . . . . . 9 ⊢ (Unit‘𝐹) = (Unit‘𝐹) | |
28 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
29 | 1, 27, 28 | isdrng 19506 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing ↔ (𝐹 ∈ Ring ∧ (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)}))) |
30 | 29 | simprbi 499 | . . . . . . 7 ⊢ (𝐹 ∈ DivRing → (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)})) |
31 | 26, 30 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)})) |
32 | 20 | fveq2d 6674 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (Unit‘𝐹) = (Unit‘(ℂfld ↾s 𝐾))) |
33 | 31, 32 | eqtr3d 2858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (𝐾 ∖ {(0g‘𝐹)}) = (Unit‘(ℂfld ↾s 𝐾))) |
34 | 25, 33 | eleqtrd 2915 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ (Unit‘(ℂfld ↾s 𝐾))) |
35 | eqid 2821 | . . . . . 6 ⊢ (Unit‘ℂfld) = (Unit‘ℂfld) | |
36 | eqid 2821 | . . . . . 6 ⊢ (Unit‘(ℂfld ↾s 𝐾)) = (Unit‘(ℂfld ↾s 𝐾)) | |
37 | eqid 2821 | . . . . . 6 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
38 | 15, 35, 36, 37 | subrgunit 19553 | . . . . 5 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → (𝑋 ∈ (Unit‘(ℂfld ↾s 𝐾)) ↔ (𝑋 ∈ (Unit‘ℂfld) ∧ 𝑋 ∈ 𝐾 ∧ ((invr‘ℂfld)‘𝑋) ∈ 𝐾))) |
39 | 6, 38 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (𝑋 ∈ (Unit‘(ℂfld ↾s 𝐾)) ↔ (𝑋 ∈ (Unit‘ℂfld) ∧ 𝑋 ∈ 𝐾 ∧ ((invr‘ℂfld)‘𝑋) ∈ 𝐾))) |
40 | 34, 39 | mpbid 234 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (𝑋 ∈ (Unit‘ℂfld) ∧ 𝑋 ∈ 𝐾 ∧ ((invr‘ℂfld)‘𝑋) ∈ 𝐾)) |
41 | 40 | simp3d 1140 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) ∈ 𝐾) |
42 | 14, 41 | eqeltrrd 2914 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 / cdiv 11297 Basecbs 16483 ↾s cress 16484 0gc0g 16713 Ringcrg 19297 Unitcui 19389 invrcinvr 19421 DivRingcdr 19502 SubRingcsubrg 19531 ℂfldccnfld 20545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-exp 13431 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-subrg 19533 df-cnfld 20546 |
This theorem is referenced by: cphreccl 23785 ipcau2 23837 |
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