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Mirrors > Home > MPE Home > Th. List > Mathboxes > sqn5i | Structured version Visualization version GIF version |
Description: The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.) |
Ref | Expression |
---|---|
sqn5i.1 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
sqn5i | ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqn5i.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
2 | 0nn0 11900 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12101 | . . . 4 ⊢ ;𝐴0 ∈ ℕ0 |
4 | 3 | nn0cni 11897 | . . 3 ⊢ ;𝐴0 ∈ ℂ |
5 | 5cn 11713 | . . 3 ⊢ 5 ∈ ℂ | |
6 | 5nn0 11905 | . . . 4 ⊢ 5 ∈ ℕ0 | |
7 | eqid 2818 | . . . 4 ⊢ ;𝐴0 = ;𝐴0 | |
8 | 5 | addid2i 10816 | . . . 4 ⊢ (0 + 5) = 5 |
9 | 1, 2, 6, 7, 8 | decaddi 12146 | . . 3 ⊢ (;𝐴0 + 5) = ;𝐴5 |
10 | eqid 2818 | . . . 4 ⊢ ;𝐴5 = ;𝐴5 | |
11 | eqid 2818 | . . . 4 ⊢ (𝐴 + 1) = (𝐴 + 1) | |
12 | 5p5e10 12157 | . . . 4 ⊢ (5 + 5) = ;10 | |
13 | 1, 6, 6, 10, 11, 12 | decaddci2 12148 | . . 3 ⊢ (;𝐴5 + 5) = ;(𝐴 + 1)0 |
14 | 4, 5, 9, 13 | sqmid3api 39047 | . 2 ⊢ (;𝐴5 · ;𝐴5) = ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) |
15 | 2nn0 11902 | . . 3 ⊢ 2 ∈ ℕ0 | |
16 | 5t5e25 12189 | . . 3 ⊢ (5 · 5) = ;25 | |
17 | peano2nn0 11925 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 1) ∈ ℕ0) | |
18 | 1, 17 | ax-mp 5 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
19 | 18, 2 | deccl 12101 | . . 3 ⊢ ;(𝐴 + 1)0 ∈ ℕ0 |
20 | 1, 18 | nn0mulcli 11923 | . . . 4 ⊢ (𝐴 · (𝐴 + 1)) ∈ ℕ0 |
21 | 1, 18, 2 | decmulnc 12153 | . . . . 5 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))(𝐴 · 0) |
22 | 1 | nn0cni 11897 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
23 | 22 | mul01i 10818 | . . . . . 6 ⊢ (𝐴 · 0) = 0 |
24 | 23 | deceq2i 12094 | . . . . 5 ⊢ ;(𝐴 · (𝐴 + 1))(𝐴 · 0) = ;(𝐴 · (𝐴 + 1))0 |
25 | 21, 24 | eqtri 2841 | . . . 4 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))0 |
26 | 2cn 11700 | . . . . 5 ⊢ 2 ∈ ℂ | |
27 | 26 | addid2i 10816 | . . . 4 ⊢ (0 + 2) = 2 |
28 | 20, 2, 15, 25, 27 | decaddi 12146 | . . 3 ⊢ ((𝐴 · ;(𝐴 + 1)0) + 2) = ;(𝐴 · (𝐴 + 1))2 |
29 | 19 | nn0cni 11897 | . . . . . 6 ⊢ ;(𝐴 + 1)0 ∈ ℂ |
30 | 29 | mul02i 10817 | . . . . 5 ⊢ (0 · ;(𝐴 + 1)0) = 0 |
31 | 30 | oveq1i 7155 | . . . 4 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = (0 + 5) |
32 | 31, 8 | eqtri 2841 | . . 3 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = 5 |
33 | 1, 2, 15, 6, 7, 16, 19, 28, 32 | decma 12137 | . 2 ⊢ ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) = ;;(𝐴 · (𝐴 + 1))25 |
34 | 14, 33 | eqtri 2841 | 1 ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 2c2 11680 5c5 11683 ℕ0cn0 11885 ;cdc 12086 |
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-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 |
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-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-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-ltxr 10668 df-sub 10860 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-dec 12087 |
This theorem is referenced by: sqn5ii 39050 |
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