Theorem pythagtriplem3 12830

Description: Lemma for pythagtrip 12846. Show that 𝐶 and 𝐵 are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem3	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵 gcd 𝐶) = 1)

Proof of Theorem pythagtriplem3

Step	Hyp	Ref	Expression
1		oveq2 6021	. . . . . . 7 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → ((𝐵↑2) gcd ((𝐴↑2) + (𝐵↑2))) = ((𝐵↑2) gcd (𝐶↑2)))
2	1	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐵↑2) gcd ((𝐴↑2) + (𝐵↑2))) = ((𝐵↑2) gcd (𝐶↑2)))
3		nnz 9488	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
4		zsqcl 10862	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
5	3, 4	syl 14	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℤ)
6	5	3ad2ant2 1043	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℤ)
7		nnz 9488	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
8		zsqcl 10862	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
9	7, 8	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℤ)
10	9	3ad2ant1 1042	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℤ)
11		gcdadd 12546	. . . . . . . . 9 ⊢ (((𝐵↑2) ∈ ℤ ∧ (𝐴↑2) ∈ ℤ) → ((𝐵↑2) gcd (𝐴↑2)) = ((𝐵↑2) gcd ((𝐴↑2) + (𝐵↑2))))
12	6, 10, 11	syl2anc 411	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵↑2) gcd (𝐴↑2)) = ((𝐵↑2) gcd ((𝐴↑2) + (𝐵↑2))))
13	6, 10	gcdcomd 12535	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵↑2) gcd (𝐴↑2)) = ((𝐴↑2) gcd (𝐵↑2)))
14	12, 13	eqtr3d 2264	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵↑2) gcd ((𝐴↑2) + (𝐵↑2))) = ((𝐴↑2) gcd (𝐵↑2)))
15	14	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐵↑2) gcd ((𝐴↑2) + (𝐵↑2))) = ((𝐴↑2) gcd (𝐵↑2)))
16	2, 15	eqtr3d 2264	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐵↑2) gcd (𝐶↑2)) = ((𝐴↑2) gcd (𝐵↑2)))
17		simpl2 1025	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 𝐵 ∈ ℕ)
18		simpl3 1026	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 𝐶 ∈ ℕ)
19		sqgcd 12590	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵 gcd 𝐶)↑2) = ((𝐵↑2) gcd (𝐶↑2)))
20	17, 18, 19	syl2anc 411	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐵 gcd 𝐶)↑2) = ((𝐵↑2) gcd (𝐶↑2)))
21		simpl1 1024	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 𝐴 ∈ ℕ)
22		sqgcd 12590	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
23	21, 17, 22	syl2anc 411	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
24	16, 20, 23	3eqtr4d 2272	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐵 gcd 𝐶)↑2) = ((𝐴 gcd 𝐵)↑2))
25	24	3adant3 1041	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐵 gcd 𝐶)↑2) = ((𝐴 gcd 𝐵)↑2))
26		simp3l 1049	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴 gcd 𝐵) = 1)
27	26	oveq1d 6028	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐴 gcd 𝐵)↑2) = (1↑2))
28	25, 27	eqtrd 2262	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐵 gcd 𝐶)↑2) = (1↑2))
29	3	3ad2ant2 1043	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℤ)
30		nnz 9488	. . . . . . 7 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
31	30	3ad2ant3 1044	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℤ)
32	29, 31	gcdcld 12529	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 gcd 𝐶) ∈ ℕ0)
33	32	nn0red 9446	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 gcd 𝐶) ∈ ℝ)
34	33	3ad2ant1 1042	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵 gcd 𝐶) ∈ ℝ)
35	32	nn0ge0d 9448	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ (𝐵 gcd 𝐶))
36	35	3ad2ant1 1042	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐵 gcd 𝐶))
37		1re 8168	. . . 4 ⊢ 1 ∈ ℝ
38		0le1 8651	. . . 4 ⊢ 0 ≤ 1
39		sq11 10864	. . . 4 ⊢ ((((𝐵 gcd 𝐶) ∈ ℝ ∧ 0 ≤ (𝐵 gcd 𝐶)) ∧ (1 ∈ ℝ ∧ 0 ≤ 1)) → (((𝐵 gcd 𝐶)↑2) = (1↑2) ↔ (𝐵 gcd 𝐶) = 1))
40	37, 38, 39	mpanr12 439	. . 3 ⊢ (((𝐵 gcd 𝐶) ∈ ℝ ∧ 0 ≤ (𝐵 gcd 𝐶)) → (((𝐵 gcd 𝐶)↑2) = (1↑2) ↔ (𝐵 gcd 𝐶) = 1))
41	34, 36, 40	syl2anc 411	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐵 gcd 𝐶)↑2) = (1↑2) ↔ (𝐵 gcd 𝐶) = 1))
42	28, 41	mpbid 147	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵 gcd 𝐶) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   class class class wbr 4086  (class class class)co 6013  ℝcr 8021  0cc0 8022  1c1 8023   + caddc 8025   ≤ cle 8205  ℕcn 9133  2c2 9184  ℤcz 9469  ↑cexp 10790   ∥ cdvds 12338   gcd cgcd 12514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7174  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515

This theorem is referenced by:  pythagtriplem4  12831