mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8565

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 8564	. . 3 $/\$ #
2	1	adantl 275	. 2 $/\$ # $/\$ $/\$ #
3		simpllr 529	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
4		simplll 528	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
5		simplrl 530	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
6		simprl 526	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
7	4, 5, 6	mulassd 7936	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 527	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 5867	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulid1d 7930	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2207	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8304	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4019	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 7933	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 7906	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8524	. . . 4 $/\$ $/\$ # #
17	14, 15, 6, 16	syl3anc 1233	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ # #
18	13, 17	mpd 13	. 2 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
19	2, 18	rexlimddv 2592	1 $/\$ # $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1348

wcel 2141

wrex 2449 class class class wbr 3987 (class class class)co 5851

cc 7765

cc0 7767

c1 7768

cmul 7772 # cap 8493

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494

This theorem is referenced by: mulap0b 8566 mulap0i 8567 mulap0d 8569 divmuldivap 8622 divdivdivap 8623 divmuleqap 8627 divadddivap 8637 conjmulap 8639 expcl2lemap 10481 expclzaplem 10493 lgsne0 13698

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