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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 10352 | . 2 ⊢ · :(ℂ × ℂ)⟶ℂ | |
2 | cnex 10353 | . . 3 ⊢ ℂ ∈ V | |
3 | 2, 2 | xpex 7240 | . 2 ⊢ (ℂ × ℂ) ∈ V |
4 | fex2 7400 | . 2 ⊢ (( · :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → · ∈ V) | |
5 | 1, 3, 2, 4 | mp3an 1534 | 1 ⊢ · ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3397 × cxp 5353 ⟶wf 6131 ℂcc 10270 · cmul 10277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-cnv 5363 df-dm 5365 df-rn 5366 df-fun 6137 df-fn 6138 df-f 6139 |
This theorem is referenced by: cnfldmul 20148 cnfldfun 20154 cnfldfunALT 20155 cnlmod4 23346 cnnvg 28105 cnnvs 28107 cncph 28246 |
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