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Mirrors > Home > MPE Home > Th. List > mulex | Structured version Visualization version GIF version |
Description: The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
mulex | ⊢ · ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-mulf 11234 | . 2 ⊢ · :(ℂ × ℂ)⟶ℂ | |
2 | cnex 11235 | . . 3 ⊢ ℂ ∈ V | |
3 | 2, 2 | xpex 7760 | . 2 ⊢ (ℂ × ℂ) ∈ V |
4 | fex2 7946 | . 2 ⊢ (( · :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → · ∈ V) | |
5 | 1, 3, 2, 4 | mp3an 1457 | 1 ⊢ · ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3461 × cxp 5679 ⟶wf 6549 ℂcc 11152 · cmul 11159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-mulf 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-xp 5687 df-rel 5688 df-cnv 5689 df-dm 5691 df-rn 5692 df-fun 6555 df-fn 6556 df-f 6557 |
This theorem is referenced by: cnfldmulOLD 21356 cnfldfunOLD 21362 cnfldfunALTOLD 21363 cnfldfunALTOLDOLD 21364 cnlmod4 25149 cnnvg 30603 cnnvs 30605 cncph 30744 |
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