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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 20263 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 20264 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
3 | eqid 2772 | . . . . 5 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
4 | eqid 2772 | . . . . 5 ⊢ (-g‘ℂfld) = (-g‘ℂfld) | |
5 | 1, 2, 3, 4 | grpsubval 17948 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(-g‘ℂfld)𝑦) = (𝑥 + ((invg‘ℂfld)‘𝑦))) |
6 | cnfldneg 20285 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → ((invg‘ℂfld)‘𝑦) = -𝑦) | |
7 | 6 | adantl 474 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((invg‘ℂfld)‘𝑦) = -𝑦) |
8 | 7 | oveq2d 6990 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + ((invg‘ℂfld)‘𝑦)) = (𝑥 + -𝑦)) |
9 | negsub 10733 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
10 | 5, 8, 9 | 3eqtrrd 2813 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥(-g‘ℂfld)𝑦)) |
11 | 10 | mpoeq3ia 7048 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
12 | subf 10686 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
13 | ffn 6341 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
15 | fnov 7096 | . . 3 ⊢ ( − Fn (ℂ × ℂ) ↔ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦))) | |
16 | 14, 15 | mpbi 222 | . 2 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) |
17 | cnring 20281 | . . . . 5 ⊢ ℂfld ∈ Ring | |
18 | ringgrp 19037 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
20 | 1, 4 | grpsubf 17977 | . . . 4 ⊢ (ℂfld ∈ Grp → (-g‘ℂfld):(ℂ × ℂ)⟶ℂ) |
21 | ffn 6341 | . . . 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 7096 | . . 3 ⊢ ((-g‘ℂfld) Fn (ℂ × ℂ) ↔ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦))) | |
24 | 22, 23 | mpbi 222 | . 2 ⊢ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
25 | 11, 16, 24 | 3eqtr4i 2806 | 1 ⊢ − = (-g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2050 × cxp 5401 Fn wfn 6180 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 ∈ cmpo 6976 ℂcc 10331 + caddc 10336 − cmin 10668 -cneg 10669 Grpcgrp 17903 invgcminusg 17904 -gcsg 17905 Ringcrg 19032 ℂfldccnfld 20259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-addf 10412 ax-mulf 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-plusg 16432 df-mulr 16433 df-starv 16434 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-minusg 17907 df-sbg 17908 df-cmn 18680 df-mgp 18975 df-ring 19034 df-cring 19035 df-cnfld 20260 |
This theorem is referenced by: zndvds 20410 resubgval 20467 cnngp 23103 cnfldtgp 23192 clmsub 23399 clmsubcl 23405 cnindmet 23481 qqhucn 30906 zringsubgval 43841 |
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