mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8551

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.)

**Assertion**
Ref	Expression
mulap0	$/\$ # $/\$ $/\$ # #

Proof of Theorem mulap0 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		recexap 8550	. . 3 $/\$ #
2	1	adantl 275	. 2 $/\$ # $/\$ $/\$ #
3		simpllr 524	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
4		simplll 523	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
5		simplrl 525	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
6		simprl 521	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
7	4, 5, 6	mulassd 7922	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 522	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 5858	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulid1d 7916	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2202	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8290	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4014	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 7919	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 7892	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8510	. . . 4 $/\$ $/\$ # #
17	14, 15, 6, 16	syl3anc 1228	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ # #
18	13, 17	mpd 13	. 2 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
19	2, 18	rexlimddv 2588	1 $/\$ # $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1343

wcel 2136

wrex 2445 class class class wbr 3982 (class class class)co 5842

cc 7751

cc0 7753

c1 7754

cmul 7758 # cap 8479

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-po 4274 df-iso 4275 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480

This theorem is referenced by: mulap0b 8552 mulap0i 8553 mulap0d 8555 divmuldivap 8608 divdivdivap 8609 divmuleqap 8613 divadddivap 8623 conjmulap 8625 expcl2lemap 10467 expclzaplem 10479 lgsne0 13579

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