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Mirrors > Home > ILE Home > Th. List > ex-exp | GIF version |
Description: Example for df-exp 10324. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-exp | ⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8806 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 5792 | . . 3 ⊢ (5↑2) = ((4 + 1)↑2) |
3 | 4cn 8822 | . . . . 5 ⊢ 4 ∈ ℂ | |
4 | binom21 10435 | . . . . 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 9018 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
7 | 4nn0 9020 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
8 | 4p1e5 8880 | . . . . 5 ⊢ (4 + 1) = 5 | |
9 | sq4e2t8 10421 | . . . . . . . 8 ⊢ (4↑2) = (2 · 8) | |
10 | 8cn 8830 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
11 | 2cn 8815 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
12 | 8t2e16 9320 | . . . . . . . . 9 ⊢ (8 · 2) = ;16 | |
13 | 10, 11, 12 | mulcomli 7797 | . . . . . . . 8 ⊢ (2 · 8) = ;16 |
14 | 9, 13 | eqtri 2161 | . . . . . . 7 ⊢ (4↑2) = ;16 |
15 | 4t2e8 8902 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
16 | 3, 11, 15 | mulcomli 7797 | . . . . . . 7 ⊢ (2 · 4) = 8 |
17 | 14, 16 | oveq12i 5794 | . . . . . 6 ⊢ ((4↑2) + (2 · 4)) = (;16 + 8) |
18 | 1nn0 9017 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
19 | 6nn0 9022 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
20 | 8nn0 9024 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
21 | eqid 2140 | . . . . . . 7 ⊢ ;16 = ;16 | |
22 | 1p1e2 8861 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
23 | 6cn 8826 | . . . . . . . 8 ⊢ 6 ∈ ℂ | |
24 | 8p6e14 9289 | . . . . . . . 8 ⊢ (8 + 6) = ;14 | |
25 | 10, 23, 24 | addcomli 7931 | . . . . . . 7 ⊢ (6 + 8) = ;14 |
26 | 18, 19, 20, 21, 22, 7, 25 | decaddci 9266 | . . . . . 6 ⊢ (;16 + 8) = ;24 |
27 | 17, 26 | eqtri 2161 | . . . . 5 ⊢ ((4↑2) + (2 · 4)) = ;24 |
28 | 6, 7, 8, 27 | decsuc 9236 | . . . 4 ⊢ (((4↑2) + (2 · 4)) + 1) = ;25 |
29 | 5, 28 | eqtri 2161 | . . 3 ⊢ ((4 + 1)↑2) = ;25 |
30 | 2, 29 | eqtri 2161 | . 2 ⊢ (5↑2) = ;25 |
31 | 3cn 8819 | . . . . 5 ⊢ 3 ∈ ℂ | |
32 | 31 | negcli 8054 | . . . 4 ⊢ -3 ∈ ℂ |
33 | 3ap0 8840 | . . . . 5 ⊢ 3 # 0 | |
34 | negap0 8416 | . . . . . 6 ⊢ (3 ∈ ℂ → (3 # 0 ↔ -3 # 0)) | |
35 | 31, 34 | ax-mp 5 | . . . . 5 ⊢ (3 # 0 ↔ -3 # 0) |
36 | 33, 35 | mpbi 144 | . . . 4 ⊢ -3 # 0 |
37 | expnegap0 10332 | . . . 4 ⊢ ((-3 ∈ ℂ ∧ -3 # 0 ∧ 2 ∈ ℕ0) → (-3↑-2) = (1 / (-3↑2))) | |
38 | 32, 36, 6, 37 | mp3an 1316 | . . 3 ⊢ (-3↑-2) = (1 / (-3↑2)) |
39 | sqneg 10383 | . . . . . 6 ⊢ (3 ∈ ℂ → (-3↑2) = (3↑2)) | |
40 | 31, 39 | ax-mp 5 | . . . . 5 ⊢ (-3↑2) = (3↑2) |
41 | sq3 10420 | . . . . 5 ⊢ (3↑2) = 9 | |
42 | 40, 41 | eqtri 2161 | . . . 4 ⊢ (-3↑2) = 9 |
43 | 42 | oveq2i 5793 | . . 3 ⊢ (1 / (-3↑2)) = (1 / 9) |
44 | 38, 43 | eqtri 2161 | . 2 ⊢ (-3↑-2) = (1 / 9) |
45 | 30, 44 | pm3.2i 270 | 1 ⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℂcc 7642 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 -cneg 7958 # cap 8367 / cdiv 8456 2c2 8795 3c3 8796 4c4 8797 5c5 8798 6c6 8799 8c8 8801 9c9 8802 ℕ0cn0 9001 ;cdc 9206 ↑cexp 10323 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-dec 9207 df-uz 9351 df-seqfrec 10250 df-exp 10324 |
This theorem is referenced by: (None) |
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