mulext - Intuitionistic Logic Explorer

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Theorem mulext 8512

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 5851. 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 519	. . . 4 $/\$ $/\$ $/\$
2		simplr 520	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 7919	. . 3 $/\$ $/\$ $/\$
4		simprl 521	. . . 4 $/\$ $/\$ $/\$
5		simprr 522	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7919	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7919	. . 3 $/\$ $/\$ $/\$
8		apcotr 8505	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1228	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8510	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1228	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8511	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8504	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 409	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 148	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 776	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wcel 2136 class class class wbr 3982 (class class class)co 5842

cc 7751

cmul 7758 # cap 8479

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480

This theorem is referenced by: mulap0r 8513 lt2msq 8781 apexp1 10631 absext 11005

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