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Mirrors > Home > MPE Home > Th. List > qabvle | Structured version Unicode version |
Description: By using induction on
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Ref | Expression |
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qrng.q |
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qabsabv.a |
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Ref | Expression |
---|---|
qabvle |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5802 |
. . . . 5
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2 | id 22 |
. . . . 5
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3 | 1, 2 | breq12d 4416 |
. . . 4
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4 | 3 | imbi2d 316 |
. . 3
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5 | fveq2 5802 |
. . . . 5
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6 | id 22 |
. . . . 5
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7 | 5, 6 | breq12d 4416 |
. . . 4
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8 | 7 | imbi2d 316 |
. . 3
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9 | fveq2 5802 |
. . . . 5
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10 | id 22 |
. . . . 5
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11 | 9, 10 | breq12d 4416 |
. . . 4
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12 | 11 | imbi2d 316 |
. . 3
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13 | fveq2 5802 |
. . . . 5
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14 | id 22 |
. . . . 5
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15 | 13, 14 | breq12d 4416 |
. . . 4
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16 | 15 | imbi2d 316 |
. . 3
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17 | qabsabv.a |
. . . . 5
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18 | qrng.q |
. . . . . 6
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19 | 18 | qrng0 23013 |
. . . . 5
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20 | 17, 19 | abv0 17049 |
. . . 4
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21 | 0le0 10526 |
. . . 4
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22 | 20, 21 | syl6eqbr 4440 |
. . 3
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23 | nn0p1nn 10734 |
. . . . . . . . . 10
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24 | 23 | ad2antrl 727 |
. . . . . . . . 9
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25 | nnq 11081 |
. . . . . . . . 9
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26 | 24, 25 | syl 16 |
. . . . . . . 8
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27 | 18 | qrngbas 23011 |
. . . . . . . . 9
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28 | 17, 27 | abvcl 17042 |
. . . . . . . 8
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29 | 26, 28 | syldan 470 |
. . . . . . 7
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30 | nn0z 10784 |
. . . . . . . . . . 11
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31 | 30 | ad2antrl 727 |
. . . . . . . . . 10
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32 | zq 11074 |
. . . . . . . . . 10
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33 | 31, 32 | syl 16 |
. . . . . . . . 9
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34 | 17, 27 | abvcl 17042 |
. . . . . . . . 9
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35 | 33, 34 | syldan 470 |
. . . . . . . 8
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36 | peano2re 9657 |
. . . . . . . 8
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37 | 35, 36 | syl 16 |
. . . . . . 7
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38 | 31 | zred 10862 |
. . . . . . . 8
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39 | peano2re 9657 |
. . . . . . . 8
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40 | 38, 39 | syl 16 |
. . . . . . 7
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41 | simpl 457 |
. . . . . . . . 9
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42 | 1z 10791 |
. . . . . . . . . 10
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43 | zq 11074 |
. . . . . . . . . 10
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44 | 42, 43 | mp1i 12 |
. . . . . . . . 9
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45 | qex 11080 |
. . . . . . . . . . 11
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46 | cnfldadd 17958 |
. . . . . . . . . . . 12
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47 | 18, 46 | ressplusg 14403 |
. . . . . . . . . . 11
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48 | 45, 47 | ax-mp 5 |
. . . . . . . . . 10
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49 | 17, 27, 48 | abvtri 17048 |
. . . . . . . . 9
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50 | 41, 33, 44, 49 | syl3anc 1219 |
. . . . . . . 8
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51 | ax-1ne0 9466 |
. . . . . . . . . . 11
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52 | 18 | qrng1 23014 |
. . . . . . . . . . . 12
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53 | 17, 52, 19 | abv1z 17050 |
. . . . . . . . . . 11
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54 | 51, 53 | mpan2 671 |
. . . . . . . . . 10
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55 | 54 | adantr 465 |
. . . . . . . . 9
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56 | 55 | oveq2d 6219 |
. . . . . . . 8
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57 | 50, 56 | breqtrd 4427 |
. . . . . . 7
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58 | 1red 9516 |
. . . . . . . 8
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59 | simprr 756 |
. . . . . . . 8
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60 | 35, 38, 58, 59 | leadd1dd 10068 |
. . . . . . 7
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61 | 29, 37, 40, 57, 60 | letrd 9643 |
. . . . . 6
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62 | 61 | expr 615 |
. . . . 5
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63 | 62 | expcom 435 |
. . . 4
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64 | 63 | a2d 26 |
. . 3
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65 | 4, 8, 12, 16, 22, 64 | nn0ind 10853 |
. 2
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66 | 65 | impcom 430 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-cnex 9453 ax-resscn 9454 ax-1cn 9455 ax-icn 9456 ax-addcl 9457 ax-addrcl 9458 ax-mulcl 9459 ax-mulrcl 9460 ax-mulcom 9461 ax-addass 9462 ax-mulass 9463 ax-distr 9464 ax-i2m1 9465 ax-1ne0 9466 ax-1rid 9467 ax-rnegex 9468 ax-rrecex 9469 ax-cnre 9470 ax-pre-lttri 9471 ax-pre-lttrn 9472 ax-pre-ltadd 9473 ax-pre-mulgt0 9474 ax-addf 9476 ax-mulf 9477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-1st 6690 df-2nd 6691 df-tpos 6858 df-recs 6945 df-rdg 6979 df-1o 7033 df-oadd 7037 df-er 7214 df-map 7329 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-pnf 9535 df-mnf 9536 df-xr 9537 df-ltxr 9538 df-le 9539 df-sub 9712 df-neg 9713 df-div 10109 df-nn 10438 df-2 10495 df-3 10496 df-4 10497 df-5 10498 df-6 10499 df-7 10500 df-8 10501 df-9 10502 df-10 10503 df-n0 10695 df-z 10762 df-dec 10871 df-uz 10977 df-q 11069 df-ico 11421 df-fz 11559 df-struct 14298 df-ndx 14299 df-slot 14300 df-base 14301 df-sets 14302 df-ress 14303 df-plusg 14374 df-mulr 14375 df-starv 14376 df-tset 14380 df-ple 14381 df-ds 14383 df-unif 14384 df-0g 14503 df-mnd 15538 df-grp 15668 df-minusg 15669 df-subg 15801 df-cmn 16404 df-mgp 16724 df-ur 16736 df-rng 16780 df-cring 16781 df-oppr 16848 df-dvdsr 16866 df-unit 16867 df-invr 16897 df-dvr 16908 df-drng 16967 df-subrg 16996 df-abv 17035 df-cnfld 17954 |
This theorem is referenced by: ostth2lem2 23026 ostth2 23029 |
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