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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recexap 8219 |
. . 3
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2 | 1 | adantl 272 |
. 2
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3 | simpllr 502 |
. . . 4
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4 | simplll 501 |
. . . . . 6
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5 | simplrl 503 |
. . . . . 6
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6 | simprl 499 |
. . . . . 6
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7 | 4, 5, 6 | mulassd 7608 |
. . . . 5
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8 | simprr 500 |
. . . . . 6
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9 | 8 | oveq2d 5706 |
. . . . 5
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10 | 4 | mulid1d 7602 |
. . . . 5
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11 | 7, 9, 10 | 3eqtrd 2131 |
. . . 4
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12 | 6 | mul02d 7967 |
. . . 4
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13 | 3, 11, 12 | 3brtr4d 3897 |
. . 3
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14 | 4, 5 | mulcld 7605 |
. . . 4
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15 | 0cnd 7578 |
. . . 4
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16 | mulext1 8186 |
. . . 4
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17 | 14, 15, 6, 16 | syl3anc 1181 |
. . 3
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18 | 13, 17 | mpd 13 |
. 2
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19 | 2, 18 | rexlimddv 2507 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 |
This theorem is referenced by: mulap0b 8221 mulap0i 8222 mulap0d 8224 divmuldivap 8276 divdivdivap 8277 divmuleqap 8281 divadddivap 8291 conjmulap 8293 expcl2lemap 10082 expclzaplem 10094 |
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