mulext - Intuitionistic Logic Explorer

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Theorem mulext 8571

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 5884. 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 527	. . . 4 $/\$ $/\$ $/\$
2		simplr 528	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 7978	. . 3 $/\$ $/\$ $/\$
4		simprl 529	. . . 4 $/\$ $/\$ $/\$
5		simprr 531	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7978	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7978	. . 3 $/\$ $/\$ $/\$
8		apcotr 8564	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1238	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8569	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1238	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8570	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8563	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 149	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 786	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708

wcel 2148 class class class wbr 4004 (class class class)co 5875

cc 7809

cmul 7816 # cap 8538

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539

This theorem is referenced by: mulap0r 8572 lt2msq 8843 apexp1 10698 absext 11072

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