mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8824

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 8823	. . 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 8193	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 531	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 6029	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulridd 8186	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2266	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8561	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4118	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 8190	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 8162	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8782	. . . 4 $/\$ $/\$ # #
17	14, 15, 6, 16	syl3anc 1271	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ # #
18	13, 17	mpd 13	. 2 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
19	2, 18	rexlimddv 2653	1 $/\$ # $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1395

wcel 2200

wrex 2509 class class class wbr 4086 (class class class)co 6013

cc 8020

cc0 8022

c1 8023

cmul 8027 # cap 8751

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-po 4391 df-iso 4392 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752

This theorem is referenced by: mulap0b 8825 mulap0i 8826 mulap0d 8828 divmuldivap 8882 divdivdivap 8883 divmuleqap 8887 divadddivap 8897 conjmulap 8899 expcl2lemap 10803 expclzaplem 10815 lgsne0 15757

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