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Mirrors > Home > ILE Home > Th. List > recextlem1 | Unicode version |
Description: Lemma for recexap 8327. (Contributed by Eric Schmidt, 23-May-2007.) |
Ref | Expression |
---|---|
recextlem1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 |
. . 3
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2 | ax-icn 7640 |
. . . . 5
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3 | mulcl 7671 |
. . . . 5
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4 | 2, 3 | mpan 418 |
. . . 4
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5 | 4 | adantl 273 |
. . 3
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6 | subcl 7884 |
. . . 4
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7 | 4, 6 | sylan2 282 |
. . 3
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8 | 1, 5, 7 | adddird 7715 |
. 2
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9 | 1, 1, 5 | subdid 8095 |
. . 3
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10 | 5, 1, 5 | subdid 8095 |
. . . 4
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11 | mulcom 7673 |
. . . . . 6
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12 | 4, 11 | sylan2 282 |
. . . . 5
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13 | ixi 8263 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | oveq1i 5738 |
. . . . . . . . 9
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15 | mulcl 7671 |
. . . . . . . . . 10
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16 | 15 | mulm1d 8091 |
. . . . . . . . 9
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17 | 14, 16 | syl5req 2160 |
. . . . . . . 8
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18 | mul4 7817 |
. . . . . . . . 9
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19 | 2, 2, 18 | mpanl12 430 |
. . . . . . . 8
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20 | 17, 19 | eqtrd 2147 |
. . . . . . 7
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21 | 20 | anidms 392 |
. . . . . 6
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22 | 21 | adantl 273 |
. . . . 5
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23 | 12, 22 | oveq12d 5746 |
. . . 4
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24 | 10, 23 | eqtr4d 2150 |
. . 3
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25 | 9, 24 | oveq12d 5746 |
. 2
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26 | mulcl 7671 |
. . . . . 6
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27 | 26 | anidms 392 |
. . . . 5
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28 | 27 | adantr 272 |
. . . 4
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29 | mulcl 7671 |
. . . . 5
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30 | 4, 29 | sylan2 282 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 15 | negcld 7983 |
. . . . . 6
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32 | 31 | anidms 392 |
. . . . 5
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33 | 32 | adantl 273 |
. . . 4
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34 | 28, 30, 33 | npncand 8020 |
. . 3
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35 | 15 | anidms 392 |
. . . 4
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36 | subneg 7934 |
. . . 4
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37 | 27, 35, 36 | syl2an 285 |
. . 3
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38 | 34, 37 | eqtrd 2147 |
. 2
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39 | 8, 25, 38 | 3eqtrd 2151 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-setind 4412 ax-resscn 7637 ax-1cn 7638 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-sub 7858 df-neg 7859 |
This theorem is referenced by: recexap 8327 |
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