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Mirrors > Home > MPE Home > Th. List > ref | Structured version Visualization version GIF version |
Description: Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
ref | ⊢ ℜ:ℂ⟶ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-re 15097 | . 2 ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) | |
2 | reval 15103 | . . 3 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) = ((𝑥 + (∗‘𝑥)) / 2)) | |
3 | recl 15107 | . . 3 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2827 | . 2 ⊢ (𝑥 ∈ ℂ → ((𝑥 + (∗‘𝑥)) / 2) ∈ ℝ) |
5 | 1, 4 | fmpti 7115 | 1 ⊢ ℜ:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 ℂcc 11144 ℝcr 11145 + caddc 11149 / cdiv 11909 2c2 12310 ∗ccj 15093 ℜcre 15094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-2 12318 df-cj 15096 df-re 15097 |
This theorem is referenced by: recn2 15595 climre 15600 rlimre 15605 caucvgr 15672 fsumre 15804 recncf 24907 cnrehmeo 24963 cnrehmeoOLD 24964 mbfdm 25640 ismbf 25642 ismbfcn 25643 mbfconst 25647 ismbfcn2 25652 mbfres 25658 mbfimaopnlem 25669 dvlip 26011 cxpcn3lem 26769 cxpcn3 26770 resqrtcn 26771 mbfresfi 37377 itgaddnc 37391 itgmulc2nc 37399 ftc1anclem5 37408 mbfres2cn 45612 |
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