mulext - Intuitionistic Logic Explorer

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Theorem mulext 8905

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 6067. 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 529	. . . 4 $/\$ $/\$ $/\$
3	1, 2	mulcld 8310	. . 3 $/\$ $/\$ $/\$
4		simprl 531	. . . 4 $/\$ $/\$ $/\$
5		simprr 533	. . . 4 $/\$ $/\$ $/\$
6	4, 5	mulcld 8310	. . 3 $/\$ $/\$ $/\$
7	4, 2	mulcld 8310	. . 3 $/\$ $/\$ $/\$
8		apcotr 8898	. . 3 $/\$ $/\$ # # $\/$ #
9	3, 6, 7, 8	syl3anc 1274	. 2 $/\$ $/\$ $/\$ # # $\/$ #
10		mulext1 8903	. . . 4 $/\$ $/\$ # #
11	1, 4, 2, 10	syl3anc 1274	. . 3 $/\$ $/\$ $/\$ # #
12		mulext2 8904	. . . . 5 $/\$ $/\$ # #
13	5, 2, 4, 12	syl3anc 1274	. . . 4 $/\$ $/\$ $/\$ # #
14		apsym 8897	. . . . 5 $/\$ # #
15	5, 2, 14	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ # #
16	13, 15	sylibd 149	. . 3 $/\$ $/\$ $/\$ # #
17	11, 16	orim12d 794	. 2 $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
18	9, 17	syld 45	1 $/\$ $/\$ $/\$ # # $\/$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wcel 2205 class class class wbr 4114 (class class class)co 6058

cc 8141

cmul 8148 # cap 8872

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-ltxr 8329 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873

This theorem is referenced by: mulap0r 8906 lt2msq 9177 apexp1 11105 absext 11773

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