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Mirrors > Home > MPE Home > Th. List > cnfldadd | Structured version Visualization version GIF version |
Description: The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21344. (Revised by GG, 27-Apr-2025.) |
Ref | Expression |
---|---|
cnfldadd | ⊢ + = (+g‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-addf 11237 | . . . 4 ⊢ + :(ℂ × ℂ)⟶ℂ | |
2 | ffn 6728 | . . . 4 ⊢ ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ + Fn (ℂ × ℂ) |
4 | fnov 7557 | . . 3 ⊢ ( + Fn (ℂ × ℂ) ↔ + = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))) | |
5 | 3, 4 | mpbi 229 | . 2 ⊢ + = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) |
6 | mpocnfldadd 21348 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld) | |
7 | 5, 6 | eqtri 2754 | 1 ⊢ + = (+g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 × cxp 5680 Fn wfn 6549 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 ℂcc 11156 + caddc 11161 +gcplusg 17266 ℂfldccnfld 21343 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-addf 11237 |
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-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-starv 17281 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-cnfld 21344 |
This theorem is referenced by: cncrng 21380 cncrngOLD 21381 cnfld0 21384 cnfldneg 21387 cnfldplusf 21388 cnfldsub 21389 cnfldmulg 21395 cnsrng 21397 cnsubmlem 21411 cnsubglem 21412 absabv 21421 cnsubrg 21424 gsumfsum 21431 regsumfsum 21432 expmhm 21433 nn0srg 21434 rge0srg 21435 zringplusg 21444 replusg 21606 regsumsupp 21618 mhpmulcl 22143 clmadd 25092 clmacl 25102 isclmp 25115 cnlmod 25158 cnncvsaddassdemo 25182 cphsqrtcl2 25205 ipcau2 25253 tdeglem3 26084 tdeglem3OLD 26085 tdeglem4 26086 tdeglem4OLD 26087 taylply2 26395 taylply2OLD 26396 efgh 26568 efabl 26577 jensenlem1 27015 jensenlem2 27016 qabvle 27654 padicabv 27659 ostth2lem2 27663 ostth3 27667 xrge0slmod 33223 zringfrac 33429 ccfldsrarelvec 33557 ccfldextdgrr 33558 2sqr3minply 33607 qqhghm 33803 qqhrhm 33804 esumpfinvallem 33907 mhphflem 42068 fsumcnsrcl 42827 rngunsnply 42834 deg1mhm 42865 amgm2d 43865 amgm3d 43866 amgm4d 43867 sge0tsms 46001 cnfldsrngadd 47539 aacllem 48549 amgmw2d 48552 |
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