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Mirrors > Home > MPE Home > Th. List > ex-exp | Structured version Visualization version GIF version |
Description: Example for df-exp 13418. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-exp | ⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11691 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 7155 | . . 3 ⊢ (5↑2) = ((4 + 1)↑2) |
3 | 4cn 11710 | . . . . 5 ⊢ 4 ∈ ℂ | |
4 | binom21 13568 | . . . . 5 ⊢ (4 ∈ ℂ → ((4 + 1)↑2) = (((4↑2) + (2 · 4)) + 1)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ((4 + 1)↑2) = (((4↑2) + (2 · 4)) + 1) |
6 | 2nn0 11902 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
7 | 4nn0 11904 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
8 | 4p1e5 11771 | . . . . 5 ⊢ (4 + 1) = 5 | |
9 | sq4e2t8 13550 | . . . . . . . 8 ⊢ (4↑2) = (2 · 8) | |
10 | 8cn 11722 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
11 | 2cn 11700 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
12 | 8t2e16 12201 | . . . . . . . . 9 ⊢ (8 · 2) = ;16 | |
13 | 10, 11, 12 | mulcomli 10638 | . . . . . . . 8 ⊢ (2 · 8) = ;16 |
14 | 9, 13 | eqtri 2841 | . . . . . . 7 ⊢ (4↑2) = ;16 |
15 | 4t2e8 11793 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
16 | 3, 11, 15 | mulcomli 10638 | . . . . . . 7 ⊢ (2 · 4) = 8 |
17 | 14, 16 | oveq12i 7157 | . . . . . 6 ⊢ ((4↑2) + (2 · 4)) = (;16 + 8) |
18 | 1nn0 11901 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | 6nn0 11906 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
20 | 8nn0 11908 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
21 | eqid 2818 | . . . . . . 7 ⊢ ;16 = ;16 | |
22 | 1p1e2 11750 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
23 | 6cn 11716 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
24 | 8p6e14 12170 | . . . . . . . 8 ⊢ (8 + 6) = ;14 | |
25 | 10, 23, 24 | addcomli 10820 | . . . . . . 7 ⊢ (6 + 8) = ;14 |
26 | 18, 19, 20, 21, 22, 7, 25 | decaddci 12147 | . . . . . 6 ⊢ (;16 + 8) = ;24 |
27 | 17, 26 | eqtri 2841 | . . . . 5 ⊢ ((4↑2) + (2 · 4)) = ;24 |
28 | 6, 7, 8, 27 | decsuc 12117 | . . . 4 ⊢ (((4↑2) + (2 · 4)) + 1) = ;25 |
29 | 5, 28 | eqtri 2841 | . . 3 ⊢ ((4 + 1)↑2) = ;25 |
30 | 2, 29 | eqtri 2841 | . 2 ⊢ (5↑2) = ;25 |
31 | 3cn 11706 | . . . . 5 ⊢ 3 ∈ ℂ | |
32 | 31 | negcli 10942 | . . . 4 ⊢ -3 ∈ ℂ |
33 | expneg 13425 | . . . 4 ⊢ ((-3 ∈ ℂ ∧ 2 ∈ ℕ0) → (-3↑-2) = (1 / (-3↑2))) | |
34 | 32, 6, 33 | mp2an 688 | . . 3 ⊢ (-3↑-2) = (1 / (-3↑2)) |
35 | sqneg 13470 | . . . . . 6 ⊢ (3 ∈ ℂ → (-3↑2) = (3↑2)) | |
36 | 31, 35 | ax-mp 5 | . . . . 5 ⊢ (-3↑2) = (3↑2) |
37 | sq3 13549 | . . . . 5 ⊢ (3↑2) = 9 | |
38 | 36, 37 | eqtri 2841 | . . . 4 ⊢ (-3↑2) = 9 |
39 | 38 | oveq2i 7156 | . . 3 ⊢ (1 / (-3↑2)) = (1 / 9) |
40 | 34, 39 | eqtri 2841 | . 2 ⊢ (-3↑-2) = (1 / 9) |
41 | 30, 40 | pm3.2i 471 | 1 ⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 1c1 10526 + caddc 10528 · cmul 10530 -cneg 10859 / cdiv 11285 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 8c8 11686 9c9 11687 ℕ0cn0 11885 ;cdc 12086 ↑cexp 13417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-seq 13358 df-exp 13418 |
This theorem is referenced by: ex-sqrt 28160 |
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