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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 21385 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 21387 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
3 | eqid 2734 | . . . . 5 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
4 | eqid 2734 | . . . . 5 ⊢ (-g‘ℂfld) = (-g‘ℂfld) | |
5 | 1, 2, 3, 4 | grpsubval 19015 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(-g‘ℂfld)𝑦) = (𝑥 + ((invg‘ℂfld)‘𝑦))) |
6 | cnfldneg 21425 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → ((invg‘ℂfld)‘𝑦) = -𝑦) | |
7 | 6 | adantl 481 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((invg‘ℂfld)‘𝑦) = -𝑦) |
8 | 7 | oveq2d 7446 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + ((invg‘ℂfld)‘𝑦)) = (𝑥 + -𝑦)) |
9 | negsub 11554 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
10 | 5, 8, 9 | 3eqtrrd 2779 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥(-g‘ℂfld)𝑦)) |
11 | 10 | mpoeq3ia 7510 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
12 | subf 11507 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
13 | ffn 6736 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
15 | fnov 7563 | . . 3 ⊢ ( − Fn (ℂ × ℂ) ↔ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦))) | |
16 | 14, 15 | mpbi 230 | . 2 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) |
17 | cnring 21420 | . . . . 5 ⊢ ℂfld ∈ Ring | |
18 | ringgrp 20255 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
20 | 1, 4 | grpsubf 19049 | . . . 4 ⊢ (ℂfld ∈ Grp → (-g‘ℂfld):(ℂ × ℂ)⟶ℂ) |
21 | ffn 6736 | . . . 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 7563 | . . 3 ⊢ ((-g‘ℂfld) Fn (ℂ × ℂ) ↔ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦))) | |
24 | 22, 23 | mpbi 230 | . 2 ⊢ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
25 | 11, 16, 24 | 3eqtr4i 2772 | 1 ⊢ − = (-g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 × cxp 5686 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ℂcc 11150 + caddc 11155 − cmin 11489 -cneg 11490 Grpcgrp 18963 invgcminusg 18964 -gcsg 18965 Ringcrg 20250 ℂfldccnfld 21381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-cmn 19814 df-mgp 20152 df-ring 20252 df-cring 20253 df-cnfld 21382 |
This theorem is referenced by: zringsub 21483 zringsubgval 21498 zndvds 21585 resubgval 21644 cnngp 24815 cnfldtgp 24906 clmsub 25126 clmsubcl 25132 cnindmet 25209 constrelextdg2 33751 2sqr3minply 33752 qqhucn 33954 |
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