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Mirrors > Home > ILE Home > Th. List > recextlem1 | Unicode version |
Description: Lemma for recexap 7743. (Contributed by Eric Schmidt, 23-May-2007.) |
Ref | Expression |
---|---|
recextlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . . 3 | |
2 | ax-icn 7071 | . . . . 5 | |
3 | mulcl 7100 | . . . . 5 | |
4 | 2, 3 | mpan 414 | . . . 4 |
5 | 4 | adantl 271 | . . 3 |
6 | subcl 7307 | . . . 4 | |
7 | 4, 6 | sylan2 280 | . . 3 |
8 | 1, 5, 7 | adddird 7144 | . 2 |
9 | 1, 1, 5 | subdid 7518 | . . 3 |
10 | 5, 1, 5 | subdid 7518 | . . . 4 |
11 | mulcom 7102 | . . . . . 6 | |
12 | 4, 11 | sylan2 280 | . . . . 5 |
13 | ixi 7683 | . . . . . . . . . 10 | |
14 | 13 | oveq1i 5542 | . . . . . . . . 9 |
15 | mulcl 7100 | . . . . . . . . . 10 | |
16 | 15 | mulm1d 7514 | . . . . . . . . 9 |
17 | 14, 16 | syl5req 2126 | . . . . . . . 8 |
18 | mul4 7240 | . . . . . . . . 9 | |
19 | 2, 2, 18 | mpanl12 426 | . . . . . . . 8 |
20 | 17, 19 | eqtrd 2113 | . . . . . . 7 |
21 | 20 | anidms 389 | . . . . . 6 |
22 | 21 | adantl 271 | . . . . 5 |
23 | 12, 22 | oveq12d 5550 | . . . 4 |
24 | 10, 23 | eqtr4d 2116 | . . 3 |
25 | 9, 24 | oveq12d 5550 | . 2 |
26 | mulcl 7100 | . . . . . 6 | |
27 | 26 | anidms 389 | . . . . 5 |
28 | 27 | adantr 270 | . . . 4 |
29 | mulcl 7100 | . . . . 5 | |
30 | 4, 29 | sylan2 280 | . . . 4 |
31 | 15 | negcld 7406 | . . . . . 6 |
32 | 31 | anidms 389 | . . . . 5 |
33 | 32 | adantl 271 | . . . 4 |
34 | 28, 30, 33 | npncand 7443 | . . 3 |
35 | 15 | anidms 389 | . . . 4 |
36 | subneg 7357 | . . . 4 | |
37 | 27, 35, 36 | syl2an 283 | . . 3 |
38 | 34, 37 | eqtrd 2113 | . 2 |
39 | 8, 25, 38 | 3eqtrd 2117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 c1 6982 ci 6983 caddc 6984 cmul 6986 cmin 7279 cneg 7280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 |
This theorem is referenced by: recexap 7743 |
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