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Mirrors > Home > MPE Home > Th. List > 4sqlem9 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16401. (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 16379 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
6 | 5 | simpld 498 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
7 | 6 | zcnd 12170 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | sqeq0 13579 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) |
10 | 9 | biimpa 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵↑2) = 0) → 𝐵 = 0) |
11 | 1, 10 | syldan 594 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = 0) |
12 | 11 | oveq2d 7187 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = (𝐴 − 0)) |
13 | 2 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ) |
14 | 13 | zcnd 12170 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ) |
15 | 14 | subid1d 11065 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 0) = 𝐴) |
16 | 12, 15 | eqtrd 2773 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = 𝐴) |
17 | 16 | oveq1d 7186 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) = (𝐴 / 𝑀)) |
18 | 5 | simprd 499 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
19 | 18 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
20 | 17, 19 | eqeltrrd 2834 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 / 𝑀) ∈ ℤ) |
21 | 3 | nnzd 12168 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
22 | 3 | nnne0d 11767 | . . . . 5 ⊢ (𝜑 → 𝑀 ≠ 0) |
23 | dvdsval2 15703 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) | |
24 | 21, 22, 2, 23 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
25 | 24 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
26 | 20, 25 | mpbird 260 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ 𝐴) |
27 | dvdssq 16009 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) | |
28 | 21, 13, 27 | syl2an2r 685 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) |
29 | 26, 28 | mpbid 235 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 class class class wbr 5031 (class class class)co 7171 ℂcc 10614 0cc0 10616 + caddc 10619 − cmin 10949 / cdiv 11376 ℕcn 11717 2c2 11772 ℤcz 12063 mod cmo 13329 ↑cexp 13522 ∥ cdvds 15700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 ax-pre-sup 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-sup 8980 df-inf 8981 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-div 11377 df-nn 11718 df-2 11780 df-3 11781 df-n0 11978 df-z 12064 df-uz 12326 df-rp 12474 df-fl 13254 df-mod 13330 df-seq 13462 df-exp 13523 df-cj 14549 df-re 14550 df-im 14551 df-sqrt 14685 df-abs 14686 df-dvds 15701 df-gcd 15939 |
This theorem is referenced by: 4sqlem16 16397 2sqlem8a 26161 |
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