mulext - Intuitionistic Logic Explorer

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Theorem mulext 8561

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 5878. 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 7968	. . 3 $/\$ $/\$ $/\$
4		simprl 529	. . . 4 $/\$ $/\$ $/\$
5		simprr 531	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7968	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7968	. . 3 $/\$ $/\$ $/\$
8		apcotr 8554	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1238	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8559	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1238	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8560	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8553	. . . . 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 4000 (class class class)co 5869

cc 7800

cmul 7807 # cap 8528

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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920

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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-ltxr 7987 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529

This theorem is referenced by: mulap0r 8562 lt2msq 8832 apexp1 10682 absext 11056

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