Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fztpval | Unicode version |
Description: Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
fztpval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9238 | . . . . 5 | |
2 | fztp 10034 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | df-3 8938 | . . . . . 6 | |
5 | 2cn 8949 | . . . . . . 7 | |
6 | ax-1cn 7867 | . . . . . . 7 | |
7 | 5, 6 | addcomi 8063 | . . . . . 6 |
8 | 4, 7 | eqtri 2191 | . . . . 5 |
9 | 8 | oveq2i 5864 | . . . 4 |
10 | tpeq3 3671 | . . . . . 6 | |
11 | 8, 10 | ax-mp 5 | . . . . 5 |
12 | df-2 8937 | . . . . . 6 | |
13 | tpeq2 3670 | . . . . . 6 | |
14 | 12, 13 | ax-mp 5 | . . . . 5 |
15 | 11, 14 | eqtri 2191 | . . . 4 |
16 | 3, 9, 15 | 3eqtr4i 2201 | . . 3 |
17 | 16 | raleqi 2669 | . 2 |
18 | 1ex 7915 | . . 3 | |
19 | 2ex 8950 | . . 3 | |
20 | 3ex 8954 | . . 3 | |
21 | fveq2 5496 | . . . 4 | |
22 | iftrue 3531 | . . . 4 | |
23 | 21, 22 | eqeq12d 2185 | . . 3 |
24 | fveq2 5496 | . . . 4 | |
25 | 1re 7919 | . . . . . . . 8 | |
26 | 1lt2 9047 | . . . . . . . 8 | |
27 | 25, 26 | gtneii 8015 | . . . . . . 7 |
28 | neeq1 2353 | . . . . . . 7 | |
29 | 27, 28 | mpbiri 167 | . . . . . 6 |
30 | ifnefalse 3537 | . . . . . 6 | |
31 | 29, 30 | syl 14 | . . . . 5 |
32 | iftrue 3531 | . . . . 5 | |
33 | 31, 32 | eqtrd 2203 | . . . 4 |
34 | 24, 33 | eqeq12d 2185 | . . 3 |
35 | fveq2 5496 | . . . 4 | |
36 | 1lt3 9049 | . . . . . . . 8 | |
37 | 25, 36 | gtneii 8015 | . . . . . . 7 |
38 | neeq1 2353 | . . . . . . 7 | |
39 | 37, 38 | mpbiri 167 | . . . . . 6 |
40 | 39, 30 | syl 14 | . . . . 5 |
41 | 2re 8948 | . . . . . . . 8 | |
42 | 2lt3 9048 | . . . . . . . 8 | |
43 | 41, 42 | gtneii 8015 | . . . . . . 7 |
44 | neeq1 2353 | . . . . . . 7 | |
45 | 43, 44 | mpbiri 167 | . . . . . 6 |
46 | ifnefalse 3537 | . . . . . 6 | |
47 | 45, 46 | syl 14 | . . . . 5 |
48 | 40, 47 | eqtrd 2203 | . . . 4 |
49 | 35, 48 | eqeq12d 2185 | . . 3 |
50 | 18, 19, 20, 23, 34, 49 | raltp 3640 | . 2 |
51 | 17, 50 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 w3a 973 wceq 1348 wcel 2141 wne 2340 wral 2448 cif 3526 ctp 3585 cfv 5198 (class class class)co 5853 c1 7775 caddc 7777 c2 8929 c3 8930 cz 9212 cfz 9965 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-tp 3591 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-2 8937 df-3 8938 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |