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Mirrors > Home > MPE Home > Th. List > cnfldsub | Structured version Visualization version GIF version |
Description: The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
cnfldsub | ⊢ − = (-g‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20800 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 20801 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
3 | eqid 2736 | . . . . 5 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
4 | eqid 2736 | . . . . 5 ⊢ (-g‘ℂfld) = (-g‘ℂfld) | |
5 | 1, 2, 3, 4 | grpsubval 18796 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(-g‘ℂfld)𝑦) = (𝑥 + ((invg‘ℂfld)‘𝑦))) |
6 | cnfldneg 20823 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → ((invg‘ℂfld)‘𝑦) = -𝑦) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((invg‘ℂfld)‘𝑦) = -𝑦) |
8 | 7 | oveq2d 7373 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + ((invg‘ℂfld)‘𝑦)) = (𝑥 + -𝑦)) |
9 | negsub 11449 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
10 | 5, 8, 9 | 3eqtrrd 2781 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥(-g‘ℂfld)𝑦)) |
11 | 10 | mpoeq3ia 7435 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
12 | subf 11403 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
13 | ffn 6668 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
15 | fnov 7487 | . . 3 ⊢ ( − Fn (ℂ × ℂ) ↔ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦))) | |
16 | 14, 15 | mpbi 229 | . 2 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) |
17 | cnring 20819 | . . . . 5 ⊢ ℂfld ∈ Ring | |
18 | ringgrp 19969 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
20 | 1, 4 | grpsubf 18826 | . . . 4 ⊢ (ℂfld ∈ Grp → (-g‘ℂfld):(ℂ × ℂ)⟶ℂ) |
21 | ffn 6668 | . . . 4 ⊢ ((-g‘ℂfld):(ℂ × ℂ)⟶ℂ → (-g‘ℂfld) Fn (ℂ × ℂ)) | |
22 | 19, 20, 21 | mp2b 10 | . . 3 ⊢ (-g‘ℂfld) Fn (ℂ × ℂ) |
23 | fnov 7487 | . . 3 ⊢ ((-g‘ℂfld) Fn (ℂ × ℂ) ↔ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦))) | |
24 | 22, 23 | mpbi 229 | . 2 ⊢ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
25 | 11, 16, 24 | 3eqtr4i 2774 | 1 ⊢ − = (-g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 × cxp 5631 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ℂcc 11049 + caddc 11054 − cmin 11385 -cneg 11386 Grpcgrp 18748 invgcminusg 18749 -gcsg 18750 Ringcrg 19964 ℂfldccnfld 20796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-cmn 19564 df-mgp 19897 df-ring 19966 df-cring 19967 df-cnfld 20797 |
This theorem is referenced by: zringsubgval 20891 zndvds 20956 resubgval 21013 cnngp 24143 cnfldtgp 24232 clmsub 24443 clmsubcl 24449 cnindmet 24526 qqhucn 32573 |
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