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Mirrors > Home > MPE Home > Th. List > 4sqlem9 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16299. (Contributed by Mario Carneiro, 15-Jul-2014.) |
Ref | Expression |
---|---|
4sqlem5.2 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
4sqlem5.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4sqlem5.4 | ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) |
4sqlem9.5 | ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = 0) |
Ref | Expression |
---|---|
4sqlem9 | ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (𝐴↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4sqlem9.5 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = 0) | |
2 | 4sqlem5.2 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
3 | 4sqlem5.3 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
4 | 4sqlem5.4 | . . . . . . . . . . . . 13 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
5 | 2, 3, 4 | 4sqlem5 16277 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
6 | 5 | simpld 497 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
7 | 6 | zcnd 12087 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | sqeq0 13485 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) |
10 | 9 | biimpa 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵↑2) = 0) → 𝐵 = 0) |
11 | 1, 10 | syldan 593 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = 0) |
12 | 11 | oveq2d 7171 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = (𝐴 − 0)) |
13 | 2 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ) |
14 | 13 | zcnd 12087 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ) |
15 | 14 | subid1d 10985 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 0) = 𝐴) |
16 | 12, 15 | eqtrd 2856 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = 𝐴) |
17 | 16 | oveq1d 7170 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) = (𝐴 / 𝑀)) |
18 | 5 | simprd 498 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
19 | 18 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
20 | 17, 19 | eqeltrrd 2914 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 / 𝑀) ∈ ℤ) |
21 | 3 | nnzd 12085 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
22 | 3 | nnne0d 11686 | . . . . 5 ⊢ (𝜑 → 𝑀 ≠ 0) |
23 | dvdsval2 15609 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) | |
24 | 21, 22, 2, 23 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
25 | 24 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
26 | 20, 25 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ 𝐴) |
27 | dvdssq 15910 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) | |
28 | 21, 13, 27 | syl2an2r 683 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) |
29 | 26, 28 | mpbid 234 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 (class class class)co 7155 ℂcc 10534 0cc0 10536 + caddc 10539 − cmin 10869 / cdiv 11296 ℕcn 11637 2c2 11691 ℤcz 11980 mod cmo 13236 ↑cexp 13428 ∥ cdvds 15606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-gcd 15843 |
This theorem is referenced by: 4sqlem16 16295 2sqlem8a 26000 |
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