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Mirrors > Home > MPE Home > Th. List > subf | Structured version Visualization version GIF version |
Description: Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Ref | Expression |
---|---|
subf | ⊢ − :(ℂ × ℂ)⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subval 10310 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) | |
2 | subcl 10318 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) ∈ ℂ) | |
3 | 1, 2 | eqeltrrd 2731 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) ∈ ℂ) |
4 | 3 | rgen2a 3006 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) ∈ ℂ |
5 | df-sub 10306 | . . 3 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) | |
6 | 5 | fmpt2 7282 | . 2 ⊢ (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) ∈ ℂ ↔ − :(ℂ × ℂ)⟶ℂ) |
7 | 4, 6 | mpbi 220 | 1 ⊢ − :(ℂ × ℂ)⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 × cxp 5141 ⟶wf 5922 ℩crio 6650 (class class class)co 6690 ℂcc 9972 + caddc 9977 − cmin 10304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 |
This theorem is referenced by: dfz2 11433 zexALT 11434 rlimsub 14418 znnen 14985 cnfldds 19804 cnfldfun 19806 cnfldfunALT 19807 cnfldsub 19822 cnmetdval 22621 cnmet 22622 cnfldms 22626 subcn 22716 cnfldcusp 23199 ovolfsf 23286 ovolctb 23304 dvlip2 23803 cnnvm 27665 mblfinlem2 33577 sblpnf 38826 fourierdlem42 40684 ovolval2lem 41178 ovolval2 41179 ovolval3 41182 |
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