Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 4sqlem8 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16514. (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 | 4sqlem5.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | 1 | nnzd 12278 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
3 | 4sqlem5.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
4 | 4sqlem5.4 | . . . . 5 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) | |
5 | 3, 1, 4 | 4sqlem5 16492 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
6 | 5 | simpld 498 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
7 | 3, 6 | zsubcld 12284 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
8 | zsqcl 13697 | . . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴↑2) ∈ ℤ) |
10 | zsqcl 13697 | . . . 4 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ) | |
11 | 6, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐵↑2) ∈ ℤ) |
12 | 9, 11 | zsubcld 12284 | . 2 ⊢ (𝜑 → ((𝐴↑2) − (𝐵↑2)) ∈ ℤ) |
13 | 5 | simprd 499 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
14 | 1 | nnne0d 11877 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 0) |
15 | dvdsval2 15815 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝑀 ∥ (𝐴 − 𝐵) ↔ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) | |
16 | 2, 14, 7, 15 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑀 ∥ (𝐴 − 𝐵) ↔ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
17 | 13, 16 | mpbird 260 | . 2 ⊢ (𝜑 → 𝑀 ∥ (𝐴 − 𝐵)) |
18 | 3, 6 | zaddcld 12283 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
19 | dvdsmul2 15837 | . . . 4 ⊢ (((𝐴 + 𝐵) ∈ ℤ ∧ (𝐴 − 𝐵) ∈ ℤ) → (𝐴 − 𝐵) ∥ ((𝐴 + 𝐵) · (𝐴 − 𝐵))) | |
20 | 18, 7, 19 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) ∥ ((𝐴 + 𝐵) · (𝐴 − 𝐵))) |
21 | 3 | zcnd 12280 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
22 | 6 | zcnd 12280 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
23 | subsq 13775 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))) | |
24 | 21, 22, 23 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))) |
25 | 20, 24 | breqtrrd 5078 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) ∥ ((𝐴↑2) − (𝐵↑2))) |
26 | 2, 7, 12, 17, 25 | dvdstrd 15853 | 1 ⊢ (𝜑 → 𝑀 ∥ ((𝐴↑2) − (𝐵↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ≠ wne 2939 class class class wbr 5050 (class class class)co 7210 ℂcc 10724 0cc0 10726 + caddc 10729 · cmul 10731 − cmin 11059 / cdiv 11486 ℕcn 11827 2c2 11882 ℤcz 12173 mod cmo 13439 ↑cexp 13632 ∥ cdvds 15812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 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 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-inf 9056 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-fl 13364 df-mod 13440 df-seq 13572 df-exp 13633 df-dvds 15813 |
This theorem is referenced by: 4sqlem14 16508 2sqlem8 26304 |
Copyright terms: Public domain | W3C validator |