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Theorem mulext 8533

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 5862. 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 524	. . . 4 $/\$ $/\$ $/\$
2		simplr 525	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 7940	. . 3 $/\$ $/\$ $/\$
4		simprl 526	. . . 4 $/\$ $/\$ $/\$
5		simprr 527	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7940	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7940	. . 3 $/\$ $/\$ $/\$
8		apcotr 8526	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1233	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8531	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1233	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8532	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1233	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8525	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 409	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 148	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 781	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wcel 2141 class class class wbr 3989 (class class class)co 5853

cc 7772

cmul 7779 # cap 8500

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501

This theorem is referenced by: mulap0r 8534 lt2msq 8802 apexp1 10652 absext 11027

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