mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8797

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 8796	. . 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 8166	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 531	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 6016	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulridd 8159	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2266	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8534	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4114	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 8163	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 8135	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8755	. . . 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 4082 (class class class)co 6000

cc 7993

cc0 7995

c1 7996

cmul 8000 # cap 8724

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 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113

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 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-opab 4145 df-id 4383 df-po 4386 df-iso 4387 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-iota 5277 df-fun 5319 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725

This theorem is referenced by: mulap0b 8798 mulap0i 8799 mulap0d 8801 divmuldivap 8855 divdivdivap 8856 divmuleqap 8860 divadddivap 8870 conjmulap 8872 expcl2lemap 10768 expclzaplem 10780 lgsne0 15711

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