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Mirrors > Home > MPE Home > Th. List > 2lgsoddprmlem3c | Structured version Visualization version GIF version |
Description: Lemma 3 for 2lgsoddprmlem3 26267. (Contributed by AV, 20-Jul-2021.) |
Ref | Expression |
---|---|
2lgsoddprmlem3c | ⊢ (((5↑2) − 1) / 8) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11879 | . . . . . . 7 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 7212 | . . . . . 6 ⊢ (5↑2) = ((4 + 1)↑2) |
3 | 4cn 11898 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
4 | binom21 13769 | . . . . . . 7 ⊢ (4 ∈ ℂ → ((4 + 1)↑2) = (((4↑2) + (2 · 4)) + 1)) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ ((4 + 1)↑2) = (((4↑2) + (2 · 4)) + 1) |
6 | 2, 5 | eqtri 2762 | . . . . 5 ⊢ (5↑2) = (((4↑2) + (2 · 4)) + 1) |
7 | 6 | oveq1i 7212 | . . . 4 ⊢ ((5↑2) − 1) = ((((4↑2) + (2 · 4)) + 1) − 1) |
8 | 3cn 11894 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 8cn 11910 | . . . . . 6 ⊢ 8 ∈ ℂ | |
10 | 8, 9 | mulcli 10823 | . . . . 5 ⊢ (3 · 8) ∈ ℂ |
11 | ax-1cn 10770 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | sq4e2t8 13751 | . . . . . . . 8 ⊢ (4↑2) = (2 · 8) | |
13 | 2cn 11888 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
14 | 4t2e8 11981 | . . . . . . . . . 10 ⊢ (4 · 2) = 8 | |
15 | 9 | mulid2i 10821 | . . . . . . . . . 10 ⊢ (1 · 8) = 8 |
16 | 14, 15 | eqtr4i 2765 | . . . . . . . . 9 ⊢ (4 · 2) = (1 · 8) |
17 | 3, 13, 16 | mulcomli 10825 | . . . . . . . 8 ⊢ (2 · 4) = (1 · 8) |
18 | 12, 17 | oveq12i 7214 | . . . . . . 7 ⊢ ((4↑2) + (2 · 4)) = ((2 · 8) + (1 · 8)) |
19 | 13, 11, 9 | adddiri 10829 | . . . . . . 7 ⊢ ((2 + 1) · 8) = ((2 · 8) + (1 · 8)) |
20 | 2p1e3 11955 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
21 | 20 | oveq1i 7212 | . . . . . . 7 ⊢ ((2 + 1) · 8) = (3 · 8) |
22 | 18, 19, 21 | 3eqtr2i 2768 | . . . . . 6 ⊢ ((4↑2) + (2 · 4)) = (3 · 8) |
23 | 22 | oveq1i 7212 | . . . . 5 ⊢ (((4↑2) + (2 · 4)) + 1) = ((3 · 8) + 1) |
24 | 10, 11, 23 | mvrraddi 11078 | . . . 4 ⊢ ((((4↑2) + (2 · 4)) + 1) − 1) = (3 · 8) |
25 | 7, 24 | eqtri 2762 | . . 3 ⊢ ((5↑2) − 1) = (3 · 8) |
26 | 25 | oveq1i 7212 | . 2 ⊢ (((5↑2) − 1) / 8) = ((3 · 8) / 8) |
27 | 0re 10818 | . . . 4 ⊢ 0 ∈ ℝ | |
28 | 8pos 11925 | . . . 4 ⊢ 0 < 8 | |
29 | 27, 28 | gtneii 10927 | . . 3 ⊢ 8 ≠ 0 |
30 | 8, 9, 29 | divcan4i 11562 | . 2 ⊢ ((3 · 8) / 8) = 3 |
31 | 26, 30 | eqtri 2762 | 1 ⊢ (((5↑2) − 1) / 8) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7202 ℂcc 10710 0cc0 10712 1c1 10713 + caddc 10715 · cmul 10717 − cmin 11045 / cdiv 11472 2c2 11868 3c3 11869 4c4 11870 5c5 11871 8c8 11874 ↑cexp 13618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-n0 12074 df-z 12160 df-uz 12422 df-seq 13558 df-exp 13619 |
This theorem is referenced by: 2lgsoddprmlem3 26267 |
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