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Mirrors > Home > MPE Home > Th. List > 2lgsoddprmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 2lgsoddprm 27262. (Contributed by AV, 19-Jul-2021.) |
Ref | Expression |
---|---|
2lgsoddprmlem1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (((𝑁↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7419 | . . . . 5 ⊢ (𝑁 = ((8 · 𝐴) + 𝐵) → (𝑁↑2) = (((8 · 𝐴) + 𝐵)↑2)) | |
2 | 1 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (𝑁↑2) = (((8 · 𝐴) + 𝐵)↑2)) |
3 | 2 | oveq1d 7427 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → ((𝑁↑2) − 1) = ((((8 · 𝐴) + 𝐵)↑2) − 1)) |
4 | 3 | oveq1d 7427 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (((𝑁↑2) − 1) / 8) = (((((8 · 𝐴) + 𝐵)↑2) − 1) / 8)) |
5 | zcn 12570 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℂ) |
7 | zcn 12570 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
8 | 7 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℂ) |
9 | 1cnd 11216 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 1 ∈ ℂ) | |
10 | 8cn 12316 | . . . . . 6 ⊢ 8 ∈ ℂ | |
11 | 8re 12315 | . . . . . . 7 ⊢ 8 ∈ ℝ | |
12 | 8pos 12331 | . . . . . . 7 ⊢ 0 < 8 | |
13 | 11, 12 | gt0ne0ii 11757 | . . . . . 6 ⊢ 8 ≠ 0 |
14 | 10, 13 | pm3.2i 470 | . . . . 5 ⊢ (8 ∈ ℂ ∧ 8 ≠ 0) |
15 | 14 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (8 ∈ ℂ ∧ 8 ≠ 0)) |
16 | mulsubdivbinom2 14229 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 1 ∈ ℂ) ∧ (8 ∈ ℂ ∧ 8 ≠ 0)) → (((((8 · 𝐴) + 𝐵)↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) | |
17 | 6, 8, 9, 15, 16 | syl31anc 1372 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((((8 · 𝐴) + 𝐵)↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) |
18 | 17 | 3adant3 1131 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (((((8 · 𝐴) + 𝐵)↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) |
19 | 4, 18 | eqtrd 2771 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (((𝑁↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 − cmin 11451 / cdiv 11878 2c2 12274 8c8 12280 ℤcz 12565 ↑cexp 14034 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-n0 12480 df-z 12566 df-uz 12830 df-seq 13974 df-exp 14035 |
This theorem is referenced by: 2lgsoddprmlem2 27255 |
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