mulext - Intuitionistic Logic Explorer

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Theorem mulext 8188

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 5699. 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 497	. . . 4 $/\$ $/\$ $/\$
2		simplr 498	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 7605	. . 3 $/\$ $/\$ $/\$
4		simprl 499	. . . 4 $/\$ $/\$ $/\$
5		simprr 500	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 7605	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 7605	. . 3 $/\$ $/\$ $/\$
8		apcotr 8181	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1181	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8186	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1181	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8187	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1181	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8180	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 404	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 148	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 738	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 667

wcel 1445 class class class wbr 3867 (class class class)co 5690

cc 7445

cmul 7452 # cap 8155

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560

This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-ltxr 7624 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156

This theorem is referenced by: mulap0r 8189 lt2msq 8444 absext 10611

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