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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 11749 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 11951 | . . . 4 ⊢ ;𝐴0 ∈ ℕ0 |
4 | 3 | nn0cni 11746 | . . 3 ⊢ ;𝐴0 ∈ ℂ |
5 | 5cn 11562 | . . 3 ⊢ 5 ∈ ℂ | |
6 | 5nn0 11754 | . . . 4 ⊢ 5 ∈ ℕ0 | |
7 | eqid 2793 | . . . 4 ⊢ ;𝐴0 = ;𝐴0 | |
8 | 5 | addid2i 10664 | . . . 4 ⊢ (0 + 5) = 5 |
9 | 1, 2, 6, 7, 8 | decaddi 11996 | . . 3 ⊢ (;𝐴0 + 5) = ;𝐴5 |
10 | eqid 2793 | . . . 4 ⊢ ;𝐴5 = ;𝐴5 | |
11 | eqid 2793 | . . . 4 ⊢ (𝐴 + 1) = (𝐴 + 1) | |
12 | 5p5e10 12008 | . . . 4 ⊢ (5 + 5) = ;10 | |
13 | 1, 6, 6, 10, 11, 12 | decaddci2 11998 | . . 3 ⊢ (;𝐴5 + 5) = ;(𝐴 + 1)0 |
14 | 4, 5, 9, 13 | sqmid3api 38641 | . 2 ⊢ (;𝐴5 · ;𝐴5) = ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) |
15 | 2nn0 11751 | . . 3 ⊢ 2 ∈ ℕ0 | |
16 | 5t5e25 12040 | . . 3 ⊢ (5 · 5) = ;25 | |
17 | peano2nn0 11774 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 1) ∈ ℕ0) | |
18 | 1, 17 | ax-mp 5 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
19 | 18, 2 | deccl 11951 | . . 3 ⊢ ;(𝐴 + 1)0 ∈ ℕ0 |
20 | 1, 18 | nn0mulcli 11772 | . . . 4 ⊢ (𝐴 · (𝐴 + 1)) ∈ ℕ0 |
21 | 1, 18, 2 | decmulnc 12004 | . . . . 5 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))(𝐴 · 0) |
22 | 1 | nn0cni 11746 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
23 | 22 | mul01i 10666 | . . . . . 6 ⊢ (𝐴 · 0) = 0 |
24 | 23 | deceq2i 11944 | . . . . 5 ⊢ ;(𝐴 · (𝐴 + 1))(𝐴 · 0) = ;(𝐴 · (𝐴 + 1))0 |
25 | 21, 24 | eqtri 2817 | . . . 4 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))0 |
26 | 2cn 11549 | . . . . 5 ⊢ 2 ∈ ℂ | |
27 | 26 | addid2i 10664 | . . . 4 ⊢ (0 + 2) = 2 |
28 | 20, 2, 15, 25, 27 | decaddi 11996 | . . 3 ⊢ ((𝐴 · ;(𝐴 + 1)0) + 2) = ;(𝐴 · (𝐴 + 1))2 |
29 | 19 | nn0cni 11746 | . . . . . 6 ⊢ ;(𝐴 + 1)0 ∈ ℂ |
30 | 29 | mul02i 10665 | . . . . 5 ⊢ (0 · ;(𝐴 + 1)0) = 0 |
31 | 30 | oveq1i 7017 | . . . 4 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = (0 + 5) |
32 | 31, 8 | eqtri 2817 | . . 3 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = 5 |
33 | 1, 2, 15, 6, 7, 16, 19, 28, 32 | decma 11987 | . 2 ⊢ ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) = ;;(𝐴 · (𝐴 + 1))25 |
34 | 14, 33 | eqtri 2817 | 1 ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 (class class class)co 7007 0cc0 10372 1c1 10373 + caddc 10375 · cmul 10377 2c2 11529 5c5 11532 ℕ0cn0 11734 ;cdc 11936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-ltxr 10515 df-sub 10708 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-dec 11937 |
This theorem is referenced by: sqn5ii 38644 |
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