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Mirrors > Home > ILE Home > Th. List > recextlem1 | Unicode version |
Description: Lemma for recexap 8541. (Contributed by Eric Schmidt, 23-May-2007.) |
Ref | Expression |
---|---|
recextlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 | |
2 | ax-icn 7839 | . . . . 5 | |
3 | mulcl 7871 | . . . . 5 | |
4 | 2, 3 | mpan 421 | . . . 4 |
5 | 4 | adantl 275 | . . 3 |
6 | subcl 8088 | . . . 4 | |
7 | 4, 6 | sylan2 284 | . . 3 |
8 | 1, 5, 7 | adddird 7915 | . 2 |
9 | 1, 1, 5 | subdid 8303 | . . 3 |
10 | 5, 1, 5 | subdid 8303 | . . . 4 |
11 | mulcom 7873 | . . . . . 6 | |
12 | 4, 11 | sylan2 284 | . . . . 5 |
13 | ixi 8472 | . . . . . . . . . 10 | |
14 | 13 | oveq1i 5846 | . . . . . . . . 9 |
15 | mulcl 7871 | . . . . . . . . . 10 | |
16 | 15 | mulm1d 8299 | . . . . . . . . 9 |
17 | 14, 16 | eqtr2id 2210 | . . . . . . . 8 |
18 | mul4 8021 | . . . . . . . . 9 | |
19 | 2, 2, 18 | mpanl12 433 | . . . . . . . 8 |
20 | 17, 19 | eqtrd 2197 | . . . . . . 7 |
21 | 20 | anidms 395 | . . . . . 6 |
22 | 21 | adantl 275 | . . . . 5 |
23 | 12, 22 | oveq12d 5854 | . . . 4 |
24 | 10, 23 | eqtr4d 2200 | . . 3 |
25 | 9, 24 | oveq12d 5854 | . 2 |
26 | mulcl 7871 | . . . . . 6 | |
27 | 26 | anidms 395 | . . . . 5 |
28 | 27 | adantr 274 | . . . 4 |
29 | mulcl 7871 | . . . . 5 | |
30 | 4, 29 | sylan2 284 | . . . 4 |
31 | 15 | negcld 8187 | . . . . . 6 |
32 | 31 | anidms 395 | . . . . 5 |
33 | 32 | adantl 275 | . . . 4 |
34 | 28, 30, 33 | npncand 8224 | . . 3 |
35 | 15 | anidms 395 | . . . 4 |
36 | subneg 8138 | . . . 4 | |
37 | 27, 35, 36 | syl2an 287 | . . 3 |
38 | 34, 37 | eqtrd 2197 | . 2 |
39 | 8, 25, 38 | 3eqtrd 2201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 (class class class)co 5836 cc 7742 c1 7745 ci 7746 caddc 7747 cmul 7749 cmin 8060 cneg 8061 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 |
This theorem is referenced by: recexap 8541 |
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