![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 4sqlem9 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16940. (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 16918 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
6 | 5 | simpld 493 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
7 | 6 | zcnd 12705 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | sqeq0 14124 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) |
10 | 9 | biimpa 475 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵↑2) = 0) → 𝐵 = 0) |
11 | 1, 10 | syldan 589 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = 0) |
12 | 11 | oveq2d 7442 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = (𝐴 − 0)) |
13 | 2 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ) |
14 | 13 | zcnd 12705 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ) |
15 | 14 | subid1d 11598 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 0) = 𝐴) |
16 | 12, 15 | eqtrd 2768 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = 𝐴) |
17 | 16 | oveq1d 7441 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) = (𝐴 / 𝑀)) |
18 | 5 | simprd 494 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
19 | 18 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
20 | 17, 19 | eqeltrrd 2830 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 / 𝑀) ∈ ℤ) |
21 | 3 | nnzd 12623 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
22 | 3 | nnne0d 12300 | . . . . 5 ⊢ (𝜑 → 𝑀 ≠ 0) |
23 | dvdsval2 16241 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) | |
24 | 21, 22, 2, 23 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
25 | 24 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
26 | 20, 25 | mpbird 256 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ 𝐴) |
27 | dvdssq 16545 | . . 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 231 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 (class class class)co 7426 ℂcc 11144 0cc0 11146 + caddc 11149 − cmin 11482 / cdiv 11909 ℕcn 12250 2c2 12305 ℤcz 12596 mod cmo 13874 ↑cexp 14066 ∥ cdvds 16238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-gcd 16477 |
This theorem is referenced by: 4sqlem16 16936 2sqlem8a 27378 |
Copyright terms: Public domain | W3C validator |