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Mirrors > Home > ILE Home > Th. List > recextlem1 | Unicode version |
Description: Lemma for recexap 8609. (Contributed by Eric Schmidt, 23-May-2007.) |
Ref | Expression |
---|---|
recextlem1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | ax-icn 7905 |
. . . . 5
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3 | mulcl 7937 |
. . . . 5
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4 | 2, 3 | mpan 424 |
. . . 4
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5 | 4 | adantl 277 |
. . 3
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6 | subcl 8155 |
. . . 4
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7 | 4, 6 | sylan2 286 |
. . 3
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8 | 1, 5, 7 | adddird 7982 |
. 2
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9 | 1, 1, 5 | subdid 8370 |
. . 3
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10 | 5, 1, 5 | subdid 8370 |
. . . 4
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11 | mulcom 7939 |
. . . . . 6
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12 | 4, 11 | sylan2 286 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | ixi 8539 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | oveq1i 5884 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | mulcl 7937 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | mulm1d 8366 |
. . . . . . . . 9
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17 | 14, 16 | eqtr2id 2223 |
. . . . . . . 8
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18 | mul4 8088 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 2, 2, 18 | mpanl12 436 |
. . . . . . . 8
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20 | 17, 19 | eqtrd 2210 |
. . . . . . 7
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21 | 20 | anidms 397 |
. . . . . 6
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22 | 21 | adantl 277 |
. . . . 5
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23 | 12, 22 | oveq12d 5892 |
. . . 4
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24 | 10, 23 | eqtr4d 2213 |
. . 3
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25 | 9, 24 | oveq12d 5892 |
. 2
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26 | mulcl 7937 |
. . . . . 6
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27 | 26 | anidms 397 |
. . . . 5
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28 | 27 | adantr 276 |
. . . 4
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29 | mulcl 7937 |
. . . . 5
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30 | 4, 29 | sylan2 286 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 15 | negcld 8254 |
. . . . . 6
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32 | 31 | anidms 397 |
. . . . 5
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33 | 32 | adantl 277 |
. . . 4
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34 | 28, 30, 33 | npncand 8291 |
. . 3
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35 | 15 | anidms 397 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | subneg 8205 |
. . . 4
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37 | 27, 35, 36 | syl2an 289 |
. . 3
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38 | 34, 37 | eqtrd 2210 |
. 2
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39 | 8, 25, 38 | 3eqtrd 2214 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8129 df-neg 8130 |
This theorem is referenced by: recexap 8609 |
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