mulext - Intuitionistic Logic Explorer

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Theorem mulext 7714

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 5541. 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 495	. . . 4 $/\$ $/\$ $/\$
2		simplr 496	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 7139	. . 3 $/\$ $/\$ $/\$
4		simprl 497	. . . 4 $/\$ $/\$ $/\$
5		simprr 498	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7139	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7139	. . 3 $/\$ $/\$ $/\$
8		apcotr 7707	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1169	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 7712	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1169	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 7713	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1169	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 7706	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 403	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 147	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 732	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 44	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103 $\/$ wo 661

wcel 1433 class class class wbr 3785 (class class class)co 5532

cc 6979

cmul 6986 # cap 7681

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682

This theorem is referenced by: mulap0r 7715 lt2msq 7964 absext 9949

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