mulext - Intuitionistic Logic Explorer

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Theorem mulext 8585

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 5897. 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 7992	. . 3 $/\$ $/\$ $/\$
4		simprl 529	. . . 4 $/\$ $/\$ $/\$
5		simprr 531	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7992	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7992	. . 3 $/\$ $/\$ $/\$
8		apcotr 8578	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1248	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8583	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1248	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8584	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1248	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8577	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 149	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 787	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wcel 2158 class class class wbr 4015 (class class class)co 5888

cc 7823

cmul 7830 # cap 8552

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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943

This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553

This theorem is referenced by: mulap0r 8586 lt2msq 8857 apexp1 10712 absext 11086

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