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Mirrors > Home > ILE Home > Th. List > recextlem1 | Unicode version |
Description: Lemma for recexap 8122. (Contributed by Eric Schmidt, 23-May-2007.) |
Ref | Expression |
---|---|
recextlem1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | ax-icn 7440 |
. . . . 5
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3 | mulcl 7469 |
. . . . 5
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4 | 2, 3 | mpan 415 |
. . . 4
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5 | 4 | adantl 271 |
. . 3
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6 | subcl 7681 |
. . . 4
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7 | 4, 6 | sylan2 280 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 1, 5, 7 | adddird 7513 |
. 2
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9 | 1, 1, 5 | subdid 7892 |
. . 3
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10 | 5, 1, 5 | subdid 7892 |
. . . 4
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11 | mulcom 7471 |
. . . . . 6
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12 | 4, 11 | sylan2 280 |
. . . . 5
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13 | ixi 8060 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | oveq1i 5662 |
. . . . . . . . 9
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15 | mulcl 7469 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | mulm1d 7888 |
. . . . . . . . 9
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17 | 14, 16 | syl5req 2133 |
. . . . . . . 8
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18 | mul4 7614 |
. . . . . . . . 9
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19 | 2, 2, 18 | mpanl12 427 |
. . . . . . . 8
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20 | 17, 19 | eqtrd 2120 |
. . . . . . 7
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21 | 20 | anidms 389 |
. . . . . 6
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22 | 21 | adantl 271 |
. . . . 5
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23 | 12, 22 | oveq12d 5670 |
. . . 4
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24 | 10, 23 | eqtr4d 2123 |
. . 3
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25 | 9, 24 | oveq12d 5670 |
. 2
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26 | mulcl 7469 |
. . . . . 6
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27 | 26 | anidms 389 |
. . . . 5
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28 | 27 | adantr 270 |
. . . 4
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29 | mulcl 7469 |
. . . . 5
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30 | 4, 29 | sylan2 280 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 15 | negcld 7780 |
. . . . . 6
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32 | 31 | anidms 389 |
. . . . 5
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33 | 32 | adantl 271 |
. . . 4
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34 | 28, 30, 33 | npncand 7817 |
. . 3
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35 | 15 | anidms 389 |
. . . 4
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36 | subneg 7731 |
. . . 4
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37 | 27, 35, 36 | syl2an 283 |
. . 3
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38 | 34, 37 | eqtrd 2120 |
. 2
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39 | 8, 25, 38 | 3eqtrd 2124 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-setind 4353 ax-resscn 7437 ax-1cn 7438 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-addcom 7445 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-i2m1 7450 ax-1rid 7452 ax-0id 7453 ax-rnegex 7454 ax-cnre 7456 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-sub 7655 df-neg 7656 |
This theorem is referenced by: recexap 8122 |
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