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Mirrors > Home > ILE Home > Th. List > mulext | Unicode version |
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 5860. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
Ref | Expression |
---|---|
mulext | # # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 524 | . . . 4 | |
2 | simplr 525 | . . . 4 | |
3 | 1, 2 | mulcld 7933 | . . 3 |
4 | simprl 526 | . . . 4 | |
5 | simprr 527 | . . . 4 | |
6 | 4, 5 | mulcld 7933 | . . 3 |
7 | 4, 2 | mulcld 7933 | . . 3 |
8 | apcotr 8519 | . . 3 # # # | |
9 | 3, 6, 7, 8 | syl3anc 1233 | . 2 # # # |
10 | mulext1 8524 | . . . 4 # # | |
11 | 1, 4, 2, 10 | syl3anc 1233 | . . 3 # # |
12 | mulext2 8525 | . . . . 5 # # | |
13 | 5, 2, 4, 12 | syl3anc 1233 | . . . 4 # # |
14 | apsym 8518 | . . . . 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 3987 (class class class)co 5851 cc 7765 cmul 7772 # cap 8493 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-ltxr 7952 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 |
This theorem is referenced by: mulap0r 8527 lt2msq 8795 apexp1 10645 absext 11020 |
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