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Mirrors > Home > ILE Home > Th. List > fzind2 | Unicode version |
Description: Induction on the integers from to inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 9298 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
Ref | Expression |
---|---|
fzind2.1 | |
fzind2.2 | |
fzind2.3 | |
fzind2.4 | |
fzind2.5 | |
fzind2.6 | ..^ |
Ref | Expression |
---|---|
fzind2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2 9943 | . . 3 | |
2 | anass 399 | . . . 4 | |
3 | df-3an 969 | . . . . 5 | |
4 | 3 | anbi1i 454 | . . . 4 |
5 | 3anass 971 | . . . . 5 | |
6 | 5 | anbi2i 453 | . . . 4 |
7 | 2, 4, 6 | 3bitr4i 211 | . . 3 |
8 | 1, 7 | bitri 183 | . 2 |
9 | fzind2.1 | . . 3 | |
10 | fzind2.2 | . . 3 | |
11 | fzind2.3 | . . 3 | |
12 | fzind2.4 | . . 3 | |
13 | eluz2 9464 | . . . 4 | |
14 | fzind2.5 | . . . 4 | |
15 | 13, 14 | sylbir 134 | . . 3 |
16 | 3anass 971 | . . . 4 | |
17 | elfzo 10075 | . . . . . . . 8 ..^ | |
18 | fzind2.6 | . . . . . . . 8 ..^ | |
19 | 17, 18 | syl6bir 163 | . . . . . . 7 |
20 | 19 | 3coml 1199 | . . . . . 6 |
21 | 20 | 3expa 1192 | . . . . 5 |
22 | 21 | impr 377 | . . . 4 |
23 | 16, 22 | sylan2b 285 | . . 3 |
24 | 9, 10, 11, 12, 15, 23 | fzind 9298 | . 2 |
25 | 8, 24 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 class class class wbr 3977 cfv 5183 (class class class)co 5837 c1 7746 caddc 7748 clt 7925 cle 7926 cz 9183 cuz 9458 cfz 9936 ..^cfzo 10068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 df-uz 9459 df-fz 9937 df-fzo 10069 |
This theorem is referenced by: exfzdc 10166 seq3clss 10393 seq3caopr3 10407 seq3f1olemp 10428 seq3id3 10433 ser3ge0 10443 prodfap0 11476 prodfrecap 11477 eulerthlemrprm 12150 eulerthlema 12151 nninfdclemlt 12347 |
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