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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 12539 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12746 | . . . 4 ⊢ ;𝐴0 ∈ ℕ0 |
4 | 3 | nn0cni 12536 | . . 3 ⊢ ;𝐴0 ∈ ℂ |
5 | 5cn 12352 | . . 3 ⊢ 5 ∈ ℂ | |
6 | 5nn0 12544 | . . . 4 ⊢ 5 ∈ ℕ0 | |
7 | eqid 2735 | . . . 4 ⊢ ;𝐴0 = ;𝐴0 | |
8 | 5 | addlidi 11447 | . . . 4 ⊢ (0 + 5) = 5 |
9 | 1, 2, 6, 7, 8 | decaddi 12791 | . . 3 ⊢ (;𝐴0 + 5) = ;𝐴5 |
10 | eqid 2735 | . . . 4 ⊢ ;𝐴5 = ;𝐴5 | |
11 | eqid 2735 | . . . 4 ⊢ (𝐴 + 1) = (𝐴 + 1) | |
12 | 5p5e10 12802 | . . . 4 ⊢ (5 + 5) = ;10 | |
13 | 1, 6, 6, 10, 11, 12 | decaddci2 12793 | . . 3 ⊢ (;𝐴5 + 5) = ;(𝐴 + 1)0 |
14 | 4, 5, 9, 13 | sqmid3api 42297 | . 2 ⊢ (;𝐴5 · ;𝐴5) = ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) |
15 | 2nn0 12541 | . . 3 ⊢ 2 ∈ ℕ0 | |
16 | 5t5e25 12834 | . . 3 ⊢ (5 · 5) = ;25 | |
17 | peano2nn0 12564 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 1) ∈ ℕ0) | |
18 | 1, 17 | ax-mp 5 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
19 | 18, 2 | deccl 12746 | . . 3 ⊢ ;(𝐴 + 1)0 ∈ ℕ0 |
20 | 1, 18 | nn0mulcli 12562 | . . . 4 ⊢ (𝐴 · (𝐴 + 1)) ∈ ℕ0 |
21 | 1, 18, 2 | decmulnc 12798 | . . . . 5 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))(𝐴 · 0) |
22 | 1 | nn0cni 12536 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
23 | 22 | mul01i 11449 | . . . . . 6 ⊢ (𝐴 · 0) = 0 |
24 | 23 | deceq2i 12739 | . . . . 5 ⊢ ;(𝐴 · (𝐴 + 1))(𝐴 · 0) = ;(𝐴 · (𝐴 + 1))0 |
25 | 21, 24 | eqtri 2763 | . . . 4 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))0 |
26 | 2cn 12339 | . . . . 5 ⊢ 2 ∈ ℂ | |
27 | 26 | addlidi 11447 | . . . 4 ⊢ (0 + 2) = 2 |
28 | 20, 2, 15, 25, 27 | decaddi 12791 | . . 3 ⊢ ((𝐴 · ;(𝐴 + 1)0) + 2) = ;(𝐴 · (𝐴 + 1))2 |
29 | 19 | nn0cni 12536 | . . . . . 6 ⊢ ;(𝐴 + 1)0 ∈ ℂ |
30 | 29 | mul02i 11448 | . . . . 5 ⊢ (0 · ;(𝐴 + 1)0) = 0 |
31 | 30 | oveq1i 7441 | . . . 4 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = (0 + 5) |
32 | 31, 8 | eqtri 2763 | . . 3 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = 5 |
33 | 1, 2, 15, 6, 7, 16, 19, 28, 32 | decma 12782 | . 2 ⊢ ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) = ;;(𝐴 · (𝐴 + 1))25 |
34 | 14, 33 | eqtri 2763 | 1 ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 2c2 12319 5c5 12322 ℕ0cn0 12524 ;cdc 12731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12732 |
This theorem is referenced by: sqn5ii 42300 |
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