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| Mirrors > Home > MPE Home > Th. List > cnfldmul | Structured version Visualization version GIF version | ||
| Description: The multiplication 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 21365. (Revised by GG, 27-Apr-2025.) |
| Ref | Expression |
|---|---|
| cnfldmul | ⊢ · = (.r‘ℂfld) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-mulf 11235 | . . . 4 ⊢ · :(ℂ × ℂ)⟶ℂ | |
| 2 | ffn 6736 | . . . 4 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ)) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ · Fn (ℂ × ℂ) |
| 4 | fnov 7564 | . . 3 ⊢ ( · Fn (ℂ × ℂ) ↔ · = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))) | |
| 5 | 3, 4 | mpbi 230 | . 2 ⊢ · = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) |
| 6 | mpocnfldmul 21371 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld) | |
| 7 | 5, 6 | eqtri 2765 | 1 ⊢ · = (.r‘ℂfld) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 · cmul 11160 .rcmulr 17298 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-cnfld 21365 |
| This theorem is referenced by: cncrngOLD 21402 cnfld1OLD 21407 cndrngOLD 21412 cnflddivOLD 21414 cnfldexp 21417 cnsrng 21418 cnsubrglemOLD 21435 absabv 21442 cnsubrg 21445 cnmsubglem 21448 expmhm 21454 nn0srg 21455 rge0srg 21456 zringmulr 21468 expghm 21486 psgnghm 21598 psgnco 21601 evpmodpmf1o 21614 remulr 21629 mdetralt 22614 clmmul 25108 clmmcl 25118 isclmp 25130 cnlmod 25173 cnncvsmulassdemo 25198 cphsubrglem 25211 cphdivcl 25216 cphabscl 25219 cphsqrtcl2 25220 cphsqrtcl3 25221 ipcau2 25268 plypf1 26251 dvply2gOLD 26327 taylply2OLD 26410 reefgim 26494 efabl 26592 efsubm 26593 amgmlem 27033 amgm 27034 wilthlem2 27112 wilthlem3 27113 dchrelbas3 27282 dchrzrhmul 27290 dchrmulcl 27293 dchrn0 27294 dchrinvcl 27297 dchrsum2 27312 sum2dchr 27318 qabvexp 27670 ostthlem2 27672 padicabv 27674 ostth2lem2 27678 ostth3 27682 xrge0slmod 33376 zringfrac 33582 ccfldsrarelvec 33721 ccfldextdgrr 33722 constrelextdg2 33788 2sqr3minply 33791 iistmd 33901 xrge0iifmhm 33938 xrge0pluscn 33939 qqhrhm 33990 cnsrexpcl 43177 cnsrplycl 43179 rngunsnply 43181 amgm2d 44211 amgm3d 44212 amgm4d 44213 cnfldsrngmul 48079 aacllem 49320 amgmlemALT 49322 amgmw2d 49323 |
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