mulap0 - Intuitionistic Logic Explorer

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Theorem mulap0 8972

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 8971	. . 3 $/\$ #
2	1	adantl 277	. 2 $/\$ # $/\$ $/\$ #
3		simpllr 540	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
4		simplll 539	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
5		simplrl 541	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
6		simprl 535	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
7	4, 5, 6	mulassd 8339	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
8		simprr 537	. . . . . 6 $/\$ # $/\$ $/\$ # $/\$ $/\$
9	8	oveq2d 6091	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
10	4	mulridd 8333	. . . . 5 $/\$ # $/\$ $/\$ # $/\$ $/\$
11	7, 9, 10	3eqtrd 2275	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
12	6	mul02d 8709	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
13	3, 11, 12	3brtr4d 4157	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
14	4, 5	mulcld 8336	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
15		0cnd 8309	. . . 4 $/\$ # $/\$ $/\$ # $/\$ $/\$
16		mulext1 8930	. . . 4 $/\$ $/\$ # #
17	14, 15, 6, 16	syl3anc 1278	. . 3 $/\$ # $/\$ $/\$ # $/\$ $/\$ # #
18	13, 17	mpd 13	. 2 $/\$ # $/\$ $/\$ # $/\$ $/\$ #
19	2, 18	rexlimddv 2673	1 $/\$ # $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1402

wcel 2209

wrex 2529 class class class wbr 4125 (class class class)co 6075

cc 8167

cc0 8169

c1 8170

cmul 8174 # cap 8899

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900

This theorem is referenced by: mulap0b 8973 mulap0i 8974 mulap0d 8976 divmuldivap 9032 divdivdivap 9033 divmuleqap 9037 divadddivap 9047 conjmulap 9049 expcl2lemap 10966 expclzaplem 10978 lgsne0 16071

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