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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 11913 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12114 | . . . 4 ⊢ ;𝐴0 ∈ ℕ0 |
4 | 3 | nn0cni 11910 | . . 3 ⊢ ;𝐴0 ∈ ℂ |
5 | 5cn 11726 | . . 3 ⊢ 5 ∈ ℂ | |
6 | 5nn0 11918 | . . . 4 ⊢ 5 ∈ ℕ0 | |
7 | eqid 2821 | . . . 4 ⊢ ;𝐴0 = ;𝐴0 | |
8 | 5 | addid2i 10828 | . . . 4 ⊢ (0 + 5) = 5 |
9 | 1, 2, 6, 7, 8 | decaddi 12159 | . . 3 ⊢ (;𝐴0 + 5) = ;𝐴5 |
10 | eqid 2821 | . . . 4 ⊢ ;𝐴5 = ;𝐴5 | |
11 | eqid 2821 | . . . 4 ⊢ (𝐴 + 1) = (𝐴 + 1) | |
12 | 5p5e10 12170 | . . . 4 ⊢ (5 + 5) = ;10 | |
13 | 1, 6, 6, 10, 11, 12 | decaddci2 12161 | . . 3 ⊢ (;𝐴5 + 5) = ;(𝐴 + 1)0 |
14 | 4, 5, 9, 13 | sqmid3api 39218 | . 2 ⊢ (;𝐴5 · ;𝐴5) = ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) |
15 | 2nn0 11915 | . . 3 ⊢ 2 ∈ ℕ0 | |
16 | 5t5e25 12202 | . . 3 ⊢ (5 · 5) = ;25 | |
17 | peano2nn0 11938 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 + 1) ∈ ℕ0) | |
18 | 1, 17 | ax-mp 5 | . . . 4 ⊢ (𝐴 + 1) ∈ ℕ0 |
19 | 18, 2 | deccl 12114 | . . 3 ⊢ ;(𝐴 + 1)0 ∈ ℕ0 |
20 | 1, 18 | nn0mulcli 11936 | . . . 4 ⊢ (𝐴 · (𝐴 + 1)) ∈ ℕ0 |
21 | 1, 18, 2 | decmulnc 12166 | . . . . 5 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))(𝐴 · 0) |
22 | 1 | nn0cni 11910 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
23 | 22 | mul01i 10830 | . . . . . 6 ⊢ (𝐴 · 0) = 0 |
24 | 23 | deceq2i 12107 | . . . . 5 ⊢ ;(𝐴 · (𝐴 + 1))(𝐴 · 0) = ;(𝐴 · (𝐴 + 1))0 |
25 | 21, 24 | eqtri 2844 | . . . 4 ⊢ (𝐴 · ;(𝐴 + 1)0) = ;(𝐴 · (𝐴 + 1))0 |
26 | 2cn 11713 | . . . . 5 ⊢ 2 ∈ ℂ | |
27 | 26 | addid2i 10828 | . . . 4 ⊢ (0 + 2) = 2 |
28 | 20, 2, 15, 25, 27 | decaddi 12159 | . . 3 ⊢ ((𝐴 · ;(𝐴 + 1)0) + 2) = ;(𝐴 · (𝐴 + 1))2 |
29 | 19 | nn0cni 11910 | . . . . . 6 ⊢ ;(𝐴 + 1)0 ∈ ℂ |
30 | 29 | mul02i 10829 | . . . . 5 ⊢ (0 · ;(𝐴 + 1)0) = 0 |
31 | 30 | oveq1i 7166 | . . . 4 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = (0 + 5) |
32 | 31, 8 | eqtri 2844 | . . 3 ⊢ ((0 · ;(𝐴 + 1)0) + 5) = 5 |
33 | 1, 2, 15, 6, 7, 16, 19, 28, 32 | decma 12150 | . 2 ⊢ ((;𝐴0 · ;(𝐴 + 1)0) + (5 · 5)) = ;;(𝐴 · (𝐴 + 1))25 |
34 | 14, 33 | eqtri 2844 | 1 ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 2c2 11693 5c5 11696 ℕ0cn0 11898 ;cdc 12099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 |
This theorem is referenced by: sqn5ii 39221 |
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