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Mirrors > Home > MPE Home > Th. List > cphreccllem | Structured version Visualization version GIF version |
Description: Lemma for cphreccl 25028. (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 25024 | . . . . . . 7 ⊢ (𝜑 → (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
5 | 4 | simp3d 1143 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (SubRing‘ℂfld)) |
6 | 5 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐾 ∈ (SubRing‘ℂfld)) |
7 | cnfldbas 21236 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
8 | 7 | subrgss 20470 | . . . . 5 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 ⊆ ℂ) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐾 ⊆ ℂ) |
10 | simp2 1136 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ 𝐾) | |
11 | 9, 10 | sseldd 3983 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ ℂ) |
12 | simp3 1137 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ≠ 0) | |
13 | cnfldinv 21264 | . . 3 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋)) | |
14 | 11, 12, 13 | syl2anc 583 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋)) |
15 | eqid 2731 | . . . . . . . . . 10 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
16 | cnfld0 21257 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
17 | 15, 16 | subrg0 20477 | . . . . . . . . 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 1141 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) |
20 | 19 | 3ad2ant1 1132 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐹 = (ℂfld ↾s 𝐾)) |
21 | 20 | fveq2d 6895 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (0g‘𝐹) = (0g‘(ℂfld ↾s 𝐾))) |
22 | 18, 21 | eqtr4d 2774 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 0 = (0g‘𝐹)) |
23 | 12, 22 | neeqtrd 3009 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ≠ (0g‘𝐹)) |
24 | eldifsn 4790 | . . . . . 6 ⊢ (𝑋 ∈ (𝐾 ∖ {(0g‘𝐹)}) ↔ (𝑋 ∈ 𝐾 ∧ 𝑋 ≠ (0g‘𝐹))) | |
25 | 10, 23, 24 | sylanbrc 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ (𝐾 ∖ {(0g‘𝐹)})) |
26 | 3 | 3ad2ant1 1132 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝐹 ∈ DivRing) |
27 | eqid 2731 | . . . . . . . . 9 ⊢ (Unit‘𝐹) = (Unit‘𝐹) | |
28 | eqid 2731 | . . . . . . . . 9 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
29 | 1, 27, 28 | isdrng 20587 | . . . . . . . 8 ⊢ (𝐹 ∈ DivRing ↔ (𝐹 ∈ Ring ∧ (Unit‘𝐹) = (𝐾 ∖ {(0g‘𝐹)}))) |
30 | 29 | simprbi 496 | . . . . . . 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 6895 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (Unit‘𝐹) = (Unit‘(ℂfld ↾s 𝐾))) |
33 | 31, 32 | eqtr3d 2773 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (𝐾 ∖ {(0g‘𝐹)}) = (Unit‘(ℂfld ↾s 𝐾))) |
34 | 25, 33 | eleqtrd 2834 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → 𝑋 ∈ (Unit‘(ℂfld ↾s 𝐾))) |
35 | eqid 2731 | . . . . . 6 ⊢ (Unit‘ℂfld) = (Unit‘ℂfld) | |
36 | eqid 2731 | . . . . . 6 ⊢ (Unit‘(ℂfld ↾s 𝐾)) = (Unit‘(ℂfld ↾s 𝐾)) | |
37 | eqid 2731 | . . . . . 6 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
38 | 15, 35, 36, 37 | subrgunit 20488 | . . . . 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 231 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (𝑋 ∈ (Unit‘ℂfld) ∧ 𝑋 ∈ 𝐾 ∧ ((invr‘ℂfld)‘𝑋) ∈ 𝐾)) |
41 | 40 | simp3d 1143 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) ∈ 𝐾) |
42 | 14, 41 | eqeltrrd 2833 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 {csn 4628 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 / cdiv 11878 Basecbs 17151 ↾s cress 17180 0gc0g 17392 Ringcrg 20134 Unitcui 20253 invrcinvr 20285 SubRingcsubrg 20465 DivRingcdr 20583 ℂfldccnfld 21232 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-seq 13974 df-exp 14035 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-subrg 20467 df-drng 20585 df-cnfld 21233 |
This theorem is referenced by: cphreccl 25028 ipcau2 25081 |
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