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Mirrors > Home > MPE Home > Th. List > 4sqlem9 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16302. (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 16280 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
6 | 5 | simpld 497 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
7 | 6 | zcnd 12091 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | sqeq0 13489 | . . . . . . . . . 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 7174 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = (𝐴 − 0)) |
13 | 2 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ) |
14 | 13 | zcnd 12091 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ) |
15 | 14 | subid1d 10988 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 0) = 𝐴) |
16 | 12, 15 | eqtrd 2858 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = 𝐴) |
17 | 16 | oveq1d 7173 | . . . 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 2916 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 / 𝑀) ∈ ℤ) |
21 | 3 | nnzd 12089 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
22 | 3 | nnne0d 11690 | . . . . 5 ⊢ (𝜑 → 𝑀 ≠ 0) |
23 | dvdsval2 15612 | . . . . 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 15913 | . . 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 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 0cc0 10539 + caddc 10542 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 ℤcz 11984 mod cmo 13240 ↑cexp 13432 ∥ cdvds 15609 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 |
This theorem is referenced by: 4sqlem16 16298 2sqlem8a 26003 |
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