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Mirrors > Home > MPE Home > Th. List > 4sqlem8 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16292. (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)) |
Ref | Expression |
---|---|
4sqlem8 | ⊢ (𝜑 → 𝑀 ∥ ((𝐴↑2) − (𝐵↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4sqlem5.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 4sqlem5.3 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
3 | 4sqlem5.4 | . . . . 5 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
4 | 1, 2, 3 | 4sqlem5 16270 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
5 | 4 | simprd 498 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
6 | 2 | nnzd 12078 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
7 | 2 | nnne0d 11679 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 0) |
8 | 4 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
9 | 1, 8 | zsubcld 12084 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
10 | dvdsval2 15602 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝑀 ∥ (𝐴 − 𝐵) ↔ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) | |
11 | 6, 7, 9, 10 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝑀 ∥ (𝐴 − 𝐵) ↔ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
12 | 5, 11 | mpbird 259 | . 2 ⊢ (𝜑 → 𝑀 ∥ (𝐴 − 𝐵)) |
13 | 1, 8 | zaddcld 12083 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
14 | dvdsmul2 15624 | . . . 4 ⊢ (((𝐴 + 𝐵) ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝐴 − 𝐵) ∥ ((𝐴 + 𝐵) · (𝐴 − 𝐵))) | |
15 | 13, 9, 14 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) ∥ ((𝐴 + 𝐵) · (𝐴 − 𝐵))) |
16 | 1 | zcnd 12080 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
17 | 8 | zcnd 12080 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
18 | subsq 13564 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))) | |
19 | 16, 17, 18 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))) |
20 | 15, 19 | breqtrrd 5085 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) ∥ ((𝐴↑2) − (𝐵↑2))) |
21 | zsqcl 13486 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
22 | 1, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴↑2) ∈ ℤ) |
23 | zsqcl 13486 | . . . . 5 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ) | |
24 | 8, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐵↑2) ∈ ℤ) |
25 | 22, 24 | zsubcld 12084 | . . 3 ⊢ (𝜑 → ((𝐴↑2) − (𝐵↑2)) ∈ ℤ) |
26 | dvdstr 15638 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ ∧ ((𝐴↑2) − (𝐵↑2)) ∈ ℤ) → ((𝑀 ∥ (𝐴 − 𝐵) ∧ (𝐴 − 𝐵) ∥ ((𝐴↑2) − (𝐵↑2))) → 𝑀 ∥ ((𝐴↑2) − (𝐵↑2)))) | |
27 | 6, 9, 25, 26 | syl3anc 1366 | . 2 ⊢ (𝜑 → ((𝑀 ∥ (𝐴 − 𝐵) ∧ (𝐴 − 𝐵) ∥ ((𝐴↑2) − (𝐵↑2))) → 𝑀 ∥ ((𝐴↑2) − (𝐵↑2)))) |
28 | 12, 20, 27 | mp2and 697 | 1 ⊢ (𝜑 → 𝑀 ∥ ((𝐴↑2) − (𝐵↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 class class class wbr 5057 (class class class)co 7148 ℂcc 10527 0cc0 10529 + caddc 10532 · cmul 10534 − cmin 10862 / cdiv 11289 ℕcn 11630 2c2 11684 ℤcz 11973 mod cmo 13229 ↑cexp 13421 ∥ cdvds 15599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-dvds 15600 |
This theorem is referenced by: 4sqlem14 16286 2sqlem8 25994 |
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