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Mirrors > Home > MPE Home > Th. List > 4sqlem9 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16893. (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 16871 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ)) |
6 | 5 | simpld 494 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
7 | 6 | zcnd 12663 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
8 | sqeq0 14081 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) = 0 ↔ 𝐵 = 0)) |
10 | 9 | biimpa 476 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐵↑2) = 0) → 𝐵 = 0) |
11 | 1, 10 | syldan 590 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = 0) |
12 | 11 | oveq2d 7417 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = (𝐴 − 0)) |
13 | 2 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ) |
14 | 13 | zcnd 12663 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ) |
15 | 14 | subid1d 11556 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 0) = 𝐴) |
16 | 12, 15 | eqtrd 2764 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 − 𝐵) = 𝐴) |
17 | 16 | oveq1d 7416 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) = (𝐴 / 𝑀)) |
18 | 5 | simprd 495 | . . . . 5 ⊢ (𝜑 → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 − 𝐵) / 𝑀) ∈ ℤ) |
20 | 17, 19 | eqeltrrd 2826 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 / 𝑀) ∈ ℤ) |
21 | 3 | nnzd 12581 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
22 | 3 | nnne0d 12258 | . . . . 5 ⊢ (𝜑 → 𝑀 ≠ 0) |
23 | dvdsval2 16196 | . . . . 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ 𝐴 ↔ (𝐴 / 𝑀) ∈ ℤ)) |
26 | 20, 25 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ 𝐴) |
27 | dvdssq 16500 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑀 ∥ 𝐴 ↔ (𝑀↑2) ∥ (𝐴↑2))) | |
28 | 21, 13, 27 | syl2an2r 682 | . 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 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5138 (class class class)co 7401 ℂcc 11103 0cc0 11105 + caddc 11108 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℤcz 12554 mod cmo 13830 ↑cexp 14023 ∥ cdvds 16193 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 |
This theorem is referenced by: 4sqlem16 16889 2sqlem8a 27262 |
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