mulext - Intuitionistic Logic Explorer

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Theorem mulext 8942

Description: Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 6094. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)

**Assertion**
Ref	Expression
mulext	$/\$ $/\$ $/\$ # # $\/$ #

**Proof of Theorem mulext**
Step	Hyp	Ref	Expression
1		simpll 531	. . . 4 $/\$ $/\$ $/\$
2		simplr 533	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 8346	. . 3 $/\$ $/\$ $/\$
4		simprl 535	. . . 4 $/\$ $/\$ $/\$
5		simprr 537	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 8346	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 8346	. . 3 $/\$ $/\$ $/\$
8		apcotr 8935	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1278	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8940	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1278	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8941	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1278	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8934	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 415	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 149	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 798	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wcel 2209 class class class wbr 4130 (class class class)co 6085

cc 8177

cmul 8184 # cap 8909

This proof depends on 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 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof 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 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-ltxr 8365 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910

This theorem is used by: mulap0r 8943 lt2msq 9216 apexp1 11156 absext 11829

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