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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ccfldextrr | Structured version Visualization version GIF version |
Description: The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.) |
Ref | Expression |
---|---|
ccfldextrr | ⊢ ℂfld/FldExtℝfld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refld 21025 | . . 3 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | rebase 21026 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
3 | 2 | oveq2i 7369 | . . 3 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s (Base‘ℝfld)) |
4 | 1, 3 | eqtri 2761 | . 2 ⊢ ℝfld = (ℂfld ↾s (Base‘ℝfld)) |
5 | resubdrg 21028 | . . . 4 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
6 | 5 | simpli 485 | . . 3 ⊢ ℝ ∈ (SubRing‘ℂfld) |
7 | 2, 6 | eqeltrri 2831 | . 2 ⊢ (Base‘ℝfld) ∈ (SubRing‘ℂfld) |
8 | cndrng 20842 | . . . 4 ⊢ ℂfld ∈ DivRing | |
9 | cncrng 20834 | . . . 4 ⊢ ℂfld ∈ CRing | |
10 | isfld 20208 | . . . 4 ⊢ (ℂfld ∈ Field ↔ (ℂfld ∈ DivRing ∧ ℂfld ∈ CRing)) | |
11 | 8, 9, 10 | mpbir2an 710 | . . 3 ⊢ ℂfld ∈ Field |
12 | refld 21039 | . . 3 ⊢ ℝfld ∈ Field | |
13 | brfldext 32393 | . . 3 ⊢ ((ℂfld ∈ Field ∧ ℝfld ∈ Field) → (ℂfld/FldExtℝfld ↔ (ℝfld = (ℂfld ↾s (Base‘ℝfld)) ∧ (Base‘ℝfld) ∈ (SubRing‘ℂfld)))) | |
14 | 11, 12, 13 | mp2an 691 | . 2 ⊢ (ℂfld/FldExtℝfld ↔ (ℝfld = (ℂfld ↾s (Base‘ℝfld)) ∧ (Base‘ℝfld) ∈ (SubRing‘ℂfld))) |
15 | 4, 7, 14 | mpbir2an 710 | 1 ⊢ ℂfld/FldExtℝfld |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 Basecbs 17088 ↾s cress 17117 CRingccrg 19970 DivRingcdr 20197 Fieldcfield 20198 SubRingcsubrg 20232 ℂfldccnfld 20812 ℝfldcrefld 21024 /FldExtcfldext 32384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-subg 18930 df-cmn 19569 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-field 20200 df-subrg 20234 df-cnfld 20813 df-refld 21025 df-fldext 32388 |
This theorem is referenced by: ccfldextdgrr 32413 |
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