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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 8641 |
. . 3
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2 | 1 | adantl 277 |
. 2
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3 | simpllr 534 |
. . . 4
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4 | simplll 533 |
. . . . . 6
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5 | simplrl 535 |
. . . . . 6
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6 | simprl 529 |
. . . . . 6
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7 | 4, 5, 6 | mulassd 8012 |
. . . . 5
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8 | simprr 531 |
. . . . . 6
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9 | 8 | oveq2d 5913 |
. . . . 5
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10 | 4 | mulridd 8005 |
. . . . 5
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11 | 7, 9, 10 | 3eqtrd 2226 |
. . . 4
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12 | 6 | mul02d 8380 |
. . . 4
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13 | 3, 11, 12 | 3brtr4d 4050 |
. . 3
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14 | 4, 5 | mulcld 8009 |
. . . 4
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15 | 0cnd 7981 |
. . . 4
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16 | mulext1 8600 |
. . . 4
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17 | 14, 15, 6, 16 | syl3anc 1249 |
. . 3
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18 | 13, 17 | mpd 13 |
. 2
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19 | 2, 18 | rexlimddv 2612 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 |
This theorem is referenced by: mulap0b 8643 mulap0i 8644 mulap0d 8646 divmuldivap 8700 divdivdivap 8701 divmuleqap 8705 divadddivap 8715 conjmulap 8717 expcl2lemap 10566 expclzaplem 10578 lgsne0 14917 |
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