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Mirrors > Home > MPE Home > Th. List > 2lgsoddprmlem3c | Structured version Visualization version GIF version |
Description: Lemma 3 for 2lgsoddprmlem3 26907. (Contributed by AV, 20-Jul-2021.) |
Ref | Expression |
---|---|
2lgsoddprmlem3c | ⊢ (((5↑2) − 1) / 8) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 12275 | . . . . . . 7 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 7416 | . . . . . 6 ⊢ (5↑2) = ((4 + 1)↑2) |
3 | 4cn 12294 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
4 | binom21 14179 | . . . . . . 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 2761 | . . . . 5 ⊢ (5↑2) = (((4↑2) + (2 · 4)) + 1) |
7 | 6 | oveq1i 7416 | . . . 4 ⊢ ((5↑2) − 1) = ((((4↑2) + (2 · 4)) + 1) − 1) |
8 | 3cn 12290 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 8cn 12306 | . . . . . 6 ⊢ 8 ∈ ℂ | |
10 | 8, 9 | mulcli 11218 | . . . . 5 ⊢ (3 · 8) ∈ ℂ |
11 | ax-1cn 11165 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | sq4e2t8 14160 | . . . . . . . 8 ⊢ (4↑2) = (2 · 8) | |
13 | 2cn 12284 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
14 | 4t2e8 12377 | . . . . . . . . . 10 ⊢ (4 · 2) = 8 | |
15 | 9 | mullidi 11216 | . . . . . . . . . 10 ⊢ (1 · 8) = 8 |
16 | 14, 15 | eqtr4i 2764 | . . . . . . . . 9 ⊢ (4 · 2) = (1 · 8) |
17 | 3, 13, 16 | mulcomli 11220 | . . . . . . . 8 ⊢ (2 · 4) = (1 · 8) |
18 | 12, 17 | oveq12i 7418 | . . . . . . 7 ⊢ ((4↑2) + (2 · 4)) = ((2 · 8) + (1 · 8)) |
19 | 13, 11, 9 | adddiri 11224 | . . . . . . 7 ⊢ ((2 + 1) · 8) = ((2 · 8) + (1 · 8)) |
20 | 2p1e3 12351 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
21 | 20 | oveq1i 7416 | . . . . . . 7 ⊢ ((2 + 1) · 8) = (3 · 8) |
22 | 18, 19, 21 | 3eqtr2i 2767 | . . . . . 6 ⊢ ((4↑2) + (2 · 4)) = (3 · 8) |
23 | 22 | oveq1i 7416 | . . . . 5 ⊢ (((4↑2) + (2 · 4)) + 1) = ((3 · 8) + 1) |
24 | 10, 11, 23 | mvrraddi 11474 | . . . 4 ⊢ ((((4↑2) + (2 · 4)) + 1) − 1) = (3 · 8) |
25 | 7, 24 | eqtri 2761 | . . 3 ⊢ ((5↑2) − 1) = (3 · 8) |
26 | 25 | oveq1i 7416 | . 2 ⊢ (((5↑2) − 1) / 8) = ((3 · 8) / 8) |
27 | 0re 11213 | . . . 4 ⊢ 0 ∈ ℝ | |
28 | 8pos 12321 | . . . 4 ⊢ 0 < 8 | |
29 | 27, 28 | gtneii 11323 | . . 3 ⊢ 8 ≠ 0 |
30 | 8, 9, 29 | divcan4i 11958 | . 2 ⊢ ((3 · 8) / 8) = 3 |
31 | 26, 30 | eqtri 2761 | 1 ⊢ (((5↑2) − 1) / 8) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7406 ℂcc 11105 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 − cmin 11441 / cdiv 11868 2c2 12264 3c3 12265 4c4 12266 5c5 12267 8c8 12270 ↑cexp 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-n0 12470 df-z 12556 df-uz 12820 df-seq 13964 df-exp 14025 |
This theorem is referenced by: 2lgsoddprmlem3 26907 |
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