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Mirrors > Home > MPE Home > Th. List > 2lgsoddprmlem3c | Structured version Visualization version GIF version |
Description: Lemma 3 for 2lgsoddprmlem3 26765. (Contributed by AV, 20-Jul-2021.) |
Ref | Expression |
---|---|
2lgsoddprmlem3c | ⊢ (((5↑2) − 1) / 8) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 12220 | . . . . . . 7 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq1i 7368 | . . . . . 6 ⊢ (5↑2) = ((4 + 1)↑2) |
3 | 4cn 12239 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
4 | binom21 14123 | . . . . . . 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 2765 | . . . . 5 ⊢ (5↑2) = (((4↑2) + (2 · 4)) + 1) |
7 | 6 | oveq1i 7368 | . . . 4 ⊢ ((5↑2) − 1) = ((((4↑2) + (2 · 4)) + 1) − 1) |
8 | 3cn 12235 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 8cn 12251 | . . . . . 6 ⊢ 8 ∈ ℂ | |
10 | 8, 9 | mulcli 11163 | . . . . 5 ⊢ (3 · 8) ∈ ℂ |
11 | ax-1cn 11110 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | sq4e2t8 14104 | . . . . . . . 8 ⊢ (4↑2) = (2 · 8) | |
13 | 2cn 12229 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
14 | 4t2e8 12322 | . . . . . . . . . 10 ⊢ (4 · 2) = 8 | |
15 | 9 | mulid2i 11161 | . . . . . . . . . 10 ⊢ (1 · 8) = 8 |
16 | 14, 15 | eqtr4i 2768 | . . . . . . . . 9 ⊢ (4 · 2) = (1 · 8) |
17 | 3, 13, 16 | mulcomli 11165 | . . . . . . . 8 ⊢ (2 · 4) = (1 · 8) |
18 | 12, 17 | oveq12i 7370 | . . . . . . 7 ⊢ ((4↑2) + (2 · 4)) = ((2 · 8) + (1 · 8)) |
19 | 13, 11, 9 | adddiri 11169 | . . . . . . 7 ⊢ ((2 + 1) · 8) = ((2 · 8) + (1 · 8)) |
20 | 2p1e3 12296 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
21 | 20 | oveq1i 7368 | . . . . . . 7 ⊢ ((2 + 1) · 8) = (3 · 8) |
22 | 18, 19, 21 | 3eqtr2i 2771 | . . . . . 6 ⊢ ((4↑2) + (2 · 4)) = (3 · 8) |
23 | 22 | oveq1i 7368 | . . . . 5 ⊢ (((4↑2) + (2 · 4)) + 1) = ((3 · 8) + 1) |
24 | 10, 11, 23 | mvrraddi 11419 | . . . 4 ⊢ ((((4↑2) + (2 · 4)) + 1) − 1) = (3 · 8) |
25 | 7, 24 | eqtri 2765 | . . 3 ⊢ ((5↑2) − 1) = (3 · 8) |
26 | 25 | oveq1i 7368 | . 2 ⊢ (((5↑2) − 1) / 8) = ((3 · 8) / 8) |
27 | 0re 11158 | . . . 4 ⊢ 0 ∈ ℝ | |
28 | 8pos 12266 | . . . 4 ⊢ 0 < 8 | |
29 | 27, 28 | gtneii 11268 | . . 3 ⊢ 8 ≠ 0 |
30 | 8, 9, 29 | divcan4i 11903 | . 2 ⊢ ((3 · 8) / 8) = 3 |
31 | 26, 30 | eqtri 2765 | 1 ⊢ (((5↑2) − 1) / 8) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7358 ℂcc 11050 0cc0 11052 1c1 11053 + caddc 11055 · cmul 11057 − cmin 11386 / cdiv 11813 2c2 12209 3c3 12210 4c4 12211 5c5 12212 8c8 12215 ↑cexp 13968 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-n0 12415 df-z 12501 df-uz 12765 df-seq 13908 df-exp 13969 |
This theorem is referenced by: 2lgsoddprmlem3 26765 |
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