mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8709

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 8708	. . 3 $/\$ #
2	1	adantl 277	. 2 $/\$ # $/\$ $/\$ #
3		simpllr 534	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
4		simplll 533	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
5		simplrl 535	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
6		simprl 529	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
7	4, 5, 6	mulassd 8078	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 531	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 5950	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulridd 8071	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2241	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8446	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4075	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 8075	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 8047	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8667	. . . 4 $/\$ $/\$ # #
17	14, 15, 6, 16	syl3anc 1249	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ # #
18	13, 17	mpd 13	. 2 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
19	2, 18	rexlimddv 2627	1 $/\$ # $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1372

wcel 2175

wrex 2484 class class class wbr 4043 (class class class)co 5934

cc 7905

cc0 7907

c1 7908

cmul 7912 # cap 8636

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-id 4338 df-po 4341 df-iso 4342 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-iota 5229 df-fun 5270 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637

This theorem is referenced by: mulap0b 8710 mulap0i 8711 mulap0d 8713 divmuldivap 8767 divdivdivap 8768 divmuleqap 8772 divadddivap 8782 conjmulap 8784 expcl2lemap 10677 expclzaplem 10689 lgsne0 15433

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