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Mirrors > Home > ILE Home > Th. List > mulap0 | Unicode version |
Description: The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.) |
Ref | Expression |
---|---|
mulap0 | # # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recexap 8407 | . . 3 # | |
2 | 1 | adantl 275 | . 2 # # |
3 | simpllr 523 | . . . 4 # # # | |
4 | simplll 522 | . . . . . 6 # # | |
5 | simplrl 524 | . . . . . 6 # # | |
6 | simprl 520 | . . . . . 6 # # | |
7 | 4, 5, 6 | mulassd 7782 | . . . . 5 # # |
8 | simprr 521 | . . . . . 6 # # | |
9 | 8 | oveq2d 5783 | . . . . 5 # # |
10 | 4 | mulid1d 7776 | . . . . 5 # # |
11 | 7, 9, 10 | 3eqtrd 2174 | . . . 4 # # |
12 | 6 | mul02d 8147 | . . . 4 # # |
13 | 3, 11, 12 | 3brtr4d 3955 | . . 3 # # # |
14 | 4, 5 | mulcld 7779 | . . . 4 # # |
15 | 0cnd 7752 | . . . 4 # # | |
16 | mulext1 8367 | . . . 4 # # | |
17 | 14, 15, 6, 16 | syl3anc 1216 | . . 3 # # # # |
18 | 13, 17 | mpd 13 | . 2 # # # |
19 | 2, 18 | rexlimddv 2552 | 1 # # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 (class class class)co 5767 cc 7611 cc0 7613 c1 7614 cmul 7618 # cap 8336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
This theorem is referenced by: mulap0b 8409 mulap0i 8410 mulap0d 8412 divmuldivap 8465 divdivdivap 8466 divmuleqap 8470 divadddivap 8480 conjmulap 8482 expcl2lemap 10298 expclzaplem 10310 |
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