Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ex-exp | Unicode version |
Description: Example for df-exp 10455. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-exp | ; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8919 | . . . 4 | |
2 | 1 | oveq1i 5852 | . . 3 |
3 | 4cn 8935 | . . . . 5 | |
4 | binom21 10567 | . . . . 5 | |
5 | 3, 4 | ax-mp 5 | . . . 4 |
6 | 2nn0 9131 | . . . . 5 | |
7 | 4nn0 9133 | . . . . 5 | |
8 | 4p1e5 8993 | . . . . 5 | |
9 | sq4e2t8 10552 | . . . . . . . 8 | |
10 | 8cn 8943 | . . . . . . . . 9 | |
11 | 2cn 8928 | . . . . . . . . 9 | |
12 | 8t2e16 9436 | . . . . . . . . 9 ; | |
13 | 10, 11, 12 | mulcomli 7906 | . . . . . . . 8 ; |
14 | 9, 13 | eqtri 2186 | . . . . . . 7 ; |
15 | 4t2e8 9015 | . . . . . . . 8 | |
16 | 3, 11, 15 | mulcomli 7906 | . . . . . . 7 |
17 | 14, 16 | oveq12i 5854 | . . . . . 6 ; |
18 | 1nn0 9130 | . . . . . . 7 | |
19 | 6nn0 9135 | . . . . . . 7 | |
20 | 8nn0 9137 | . . . . . . 7 | |
21 | eqid 2165 | . . . . . . 7 ; ; | |
22 | 1p1e2 8974 | . . . . . . 7 | |
23 | 6cn 8939 | . . . . . . . 8 | |
24 | 8p6e14 9405 | . . . . . . . 8 ; | |
25 | 10, 23, 24 | addcomli 8043 | . . . . . . 7 ; |
26 | 18, 19, 20, 21, 22, 7, 25 | decaddci 9382 | . . . . . 6 ; ; |
27 | 17, 26 | eqtri 2186 | . . . . 5 ; |
28 | 6, 7, 8, 27 | decsuc 9352 | . . . 4 ; |
29 | 5, 28 | eqtri 2186 | . . 3 ; |
30 | 2, 29 | eqtri 2186 | . 2 ; |
31 | 3cn 8932 | . . . . 5 | |
32 | 31 | negcli 8166 | . . . 4 |
33 | 3ap0 8953 | . . . . 5 # | |
34 | negap0 8528 | . . . . . 6 # # | |
35 | 31, 34 | ax-mp 5 | . . . . 5 # # |
36 | 33, 35 | mpbi 144 | . . . 4 # |
37 | expnegap0 10463 | . . . 4 # | |
38 | 32, 36, 6, 37 | mp3an 1327 | . . 3 |
39 | sqneg 10514 | . . . . . 6 | |
40 | 31, 39 | ax-mp 5 | . . . . 5 |
41 | sq3 10551 | . . . . 5 | |
42 | 40, 41 | eqtri 2186 | . . . 4 |
43 | 42 | oveq2i 5853 | . . 3 |
44 | 38, 43 | eqtri 2186 | . 2 |
45 | 30, 44 | pm3.2i 270 | 1 ; |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1343 wcel 2136 class class class wbr 3982 (class class class)co 5842 cc 7751 cc0 7753 c1 7754 caddc 7756 cmul 7758 cneg 8070 # cap 8479 cdiv 8568 c2 8908 c3 8909 c4 8910 c5 8911 c6 8912 c8 8914 c9 8915 cn0 9114 ;cdc 9322 cexp 10454 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-z 9192 df-dec 9323 df-uz 9467 df-seqfrec 10381 df-exp 10455 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |