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Mirrors > Home > ILE Home > Th. List > mulcanenq | Unicode version |
Description: Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) |
Ref | Expression |
---|---|
mulcanenq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 992 | . . 3 | |
2 | simp2 993 | . . 3 | |
3 | simp3 994 | . . 3 | |
4 | mulcompig 7286 | . . . 4 | |
5 | 4 | adantl 275 | . . 3 |
6 | mulasspig 7287 | . . . 4 | |
7 | 6 | adantl 275 | . . 3 |
8 | 1, 2, 3, 5, 7 | caov32d 6031 | . 2 |
9 | mulclpi 7283 | . . . . . 6 | |
10 | mulclpi 7283 | . . . . . 6 | |
11 | 9, 10 | anim12i 336 | . . . . 5 |
12 | simpr 109 | . . . . . 6 | |
13 | 12 | an4s 583 | . . . . 5 |
14 | 11, 13 | jca 304 | . . . 4 |
15 | 14 | 3impdi 1288 | . . 3 |
16 | enqbreq 7311 | . . 3 | |
17 | 15, 16 | syl 14 | . 2 |
18 | 8, 17 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 cop 3584 class class class wbr 3987 (class class class)co 5851 cnpi 7227 cmi 7229 ceq 7234 |
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-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 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-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-oadd 6397 df-omul 6398 df-ni 7259 df-mi 7261 df-enq 7302 |
This theorem is referenced by: mulcanenqec 7341 |
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