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Mirrors > Home > ILE Home > Th. List > dvdssqlem | GIF version |
Description: Lemma for dvdssq 11949. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
dvdssqlem | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9201 | . . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
2 | nnz 9201 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
3 | dvdssqim 11942 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝑀↑2) ∥ (𝑁↑2))) | |
4 | 1, 2, 3 | syl2an 287 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 → (𝑀↑2) ∥ (𝑁↑2))) |
5 | sqgcd 11947 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) = ((𝑀↑2) gcd (𝑁↑2))) | |
6 | 5 | adantr 274 | . . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → ((𝑀 gcd 𝑁)↑2) = ((𝑀↑2) gcd (𝑁↑2))) |
7 | nnsqcl 10514 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀↑2) ∈ ℕ) | |
8 | nnsqcl 10514 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℕ) | |
9 | gcdeq 11941 | . . . . . . . 8 ⊢ (((𝑀↑2) ∈ ℕ ∧ (𝑁↑2) ∈ ℕ) → (((𝑀↑2) gcd (𝑁↑2)) = (𝑀↑2) ↔ (𝑀↑2) ∥ (𝑁↑2))) | |
10 | 7, 8, 9 | syl2an 287 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀↑2) gcd (𝑁↑2)) = (𝑀↑2) ↔ (𝑀↑2) ∥ (𝑁↑2))) |
11 | 10 | biimpar 295 | . . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → ((𝑀↑2) gcd (𝑁↑2)) = (𝑀↑2)) |
12 | 6, 11 | eqtrd 2197 | . . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → ((𝑀 gcd 𝑁)↑2) = (𝑀↑2)) |
13 | gcdcl 11884 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ0) | |
14 | 1, 2, 13 | syl2an 287 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℕ0) |
15 | 14 | nn0red 9159 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℝ) |
16 | 14 | nn0ge0d 9161 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝑀 gcd 𝑁)) |
17 | nnre 8855 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
18 | 17 | adantr 274 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ) |
19 | nnnn0 9112 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0) | |
20 | 19 | nn0ge0d 9161 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀) |
21 | 20 | adantr 274 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑀) |
22 | sq11 10517 | . . . . . . 7 ⊢ ((((𝑀 gcd 𝑁) ∈ ℝ ∧ 0 ≤ (𝑀 gcd 𝑁)) ∧ (𝑀 ∈ ℝ ∧ 0 ≤ 𝑀)) → (((𝑀 gcd 𝑁)↑2) = (𝑀↑2) ↔ (𝑀 gcd 𝑁) = 𝑀)) | |
23 | 15, 16, 18, 21, 22 | syl22anc 1228 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 gcd 𝑁)↑2) = (𝑀↑2) ↔ (𝑀 gcd 𝑁) = 𝑀)) |
24 | 23 | adantr 274 | . . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → (((𝑀 gcd 𝑁)↑2) = (𝑀↑2) ↔ (𝑀 gcd 𝑁) = 𝑀)) |
25 | 12, 24 | mpbid 146 | . . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → (𝑀 gcd 𝑁) = 𝑀) |
26 | gcddvds 11881 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) | |
27 | 1, 2, 26 | syl2an 287 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
28 | 27 | adantr 274 | . . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
29 | 28 | simprd 113 | . . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → (𝑀 gcd 𝑁) ∥ 𝑁) |
30 | 25, 29 | eqbrtrrd 4000 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀↑2) ∥ (𝑁↑2)) → 𝑀 ∥ 𝑁) |
31 | 30 | ex 114 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀↑2) ∥ (𝑁↑2) → 𝑀 ∥ 𝑁)) |
32 | 4, 31 | impbid 128 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 ℝcr 7743 0cc0 7744 ≤ cle 7925 ℕcn 8848 2c2 8899 ℕ0cn0 9105 ℤcz 9182 ↑cexp 10444 ∥ cdvds 11713 gcd cgcd 11860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861 |
This theorem is referenced by: dvdssq 11949 |
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