mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8698

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 8697	. . 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 8067	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 531	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 5941	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulridd 8060	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2233	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8435	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4066	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 8064	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 8036	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8656	. . . 4 $/\$ $/\$ # #
17	14, 15, 6, 16	syl3anc 1249	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ # #
18	13, 17	mpd 13	. 2 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
19	2, 18	rexlimddv 2619	1 $/\$ # $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1364

wcel 2167

wrex 2476 class class class wbr 4034 (class class class)co 5925

cc 7894

cc0 7896

c1 7897

cmul 7901 # cap 8625

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626

This theorem is referenced by: mulap0b 8699 mulap0i 8700 mulap0d 8702 divmuldivap 8756 divdivdivap 8757 divmuleqap 8761 divadddivap 8771 conjmulap 8773 expcl2lemap 10660 expclzaplem 10672 lgsne0 15363

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