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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 11597 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 11798 | . . . 4 ⊢ ;𝐴0 ∈ ℕ0 |
4 | 3 | nn0cni 11593 | . . 3 ⊢ ;𝐴0 ∈ ℂ |
5 | 5cn 11403 | . . 3 ⊢ 5 ∈ ℂ | |
6 | 5nn0 11602 | . . . 4 ⊢ 5 ∈ ℕ0 | |
7 | eqid 2799 | . . . 4 ⊢ ;𝐴0 = ;𝐴0 | |
8 | 5 | addid2i 10514 | . . . 4 ⊢ (0 + 5) = 5 |
9 | 1, 2, 6, 7, 8 | decaddi 11844 | . . 3 ⊢ (;𝐴0 + 5) = ;𝐴5 |
10 | eqid 2799 | . . . 4 ⊢ ;𝐴5 = ;𝐴5 | |
11 | eqid 2799 | . . . 4 ⊢ (𝐴 + 1) = (𝐴 + 1) | |
12 | 5p5e10 11856 | . . . 4 ⊢ (5 + 5) = ;10 | |
13 | 1, 6, 6, 10, 11, 12 | decaddci2 11846 | . . 3 ⊢ (;𝐴5 + 5) = ;(𝐴 + 1)0 |
14 | 4, 5, 9, 13 | sqmid3api 37994 | . 2 ⊢ (;𝐴5 · ;𝐴5) = ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) |
15 | 2nn0 11599 | . . 3 ⊢ 2 ∈ ℕ0 | |
16 | 5t5e25 11888 | . . 3 ⊢ (5 · 5) = ;25 | |
17 | peano2nn0 11622 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 1) ∈ ℕ0) | |
18 | 1, 17 | ax-mp 5 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
19 | 18, 2 | deccl 11798 | . . 3 ⊢ ;(𝐴 + 1)0 ∈ ℕ0 |
20 | 1, 18 | nn0mulcli 11620 | . . . 4 ⊢ (𝐴 · (𝐴 + 1)) ∈ ℕ0 |
21 | 1, 18, 2 | decmulnc 11852 | . . . . 5 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))(𝐴 · 0) |
22 | 1 | nn0cni 11593 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
23 | 22 | mul01i 10516 | . . . . . 6 ⊢ (𝐴 · 0) = 0 |
24 | 23 | deceq2i 11791 | . . . . 5 ⊢ ;(𝐴 · (𝐴 + 1))(𝐴 · 0) = ;(𝐴 · (𝐴 + 1))0 |
25 | 21, 24 | eqtri 2821 | . . . 4 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))0 |
26 | 2cn 11388 | . . . . 5 ⊢ 2 ∈ ℂ | |
27 | 26 | addid2i 10514 | . . . 4 ⊢ (0 + 2) = 2 |
28 | 20, 2, 15, 25, 27 | decaddi 11844 | . . 3 ⊢ ((𝐴 · ;(𝐴 + 1)0) + 2) = ;(𝐴 · (𝐴 + 1))2 |
29 | 19 | nn0cni 11593 | . . . . . 6 ⊢ ;(𝐴 + 1)0 ∈ ℂ |
30 | 29 | mul02i 10515 | . . . . 5 ⊢ (0 · ;(𝐴 + 1)0) = 0 |
31 | 30 | oveq1i 6888 | . . . 4 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = (0 + 5) |
32 | 31, 8 | eqtri 2821 | . . 3 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = 5 |
33 | 1, 2, 15, 6, 7, 16, 19, 28, 32 | decma 11835 | . 2 ⊢ ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) = ;;(𝐴 · (𝐴 + 1))25 |
34 | 14, 33 | eqtri 2821 | 1 ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 (class class class)co 6878 0cc0 10224 1c1 10225 + caddc 10227 · cmul 10229 2c2 11368 5c5 11371 ℕ0cn0 11580 ;cdc 11783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-dec 11784 |
This theorem is referenced by: sqn5ii 37997 |
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