Theorem 1qec 3862

Description: The equivalence class of ratio 1.

Hypothesis

Ref	Expression
1qec.1	⊢ A ∈ V

Assertion

Ref	Expression
1qec	⊢ (A ∈ N → 1Q = [⟨A, A⟩] ~Q )

Proof of Theorem 1qec

Step	Hyp	Ref	Expression
1		1pi 3805	. . . 4 ⊢ 1o ∈ N
2		1qec.1	. . . . . 6 ⊢ A ∈ V
3	1	elisseti 1355	. . . . . 6 ⊢ 1o ∈ V
4	2, 3, 3	distrpqlem 3860	. . . . 5 ⊢ ((A ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → [⟨(A ·N 1o), (A ·N 1o)⟩] ~Q = [⟨1o, 1o⟩] ~Q )
5	1, 4	mp3an2 640	. . . 4 ⊢ ((A ∈ N ∧ 1o ∈ N) → [⟨(A ·N 1o), (A ·N 1o)⟩] ~Q = [⟨1o, 1o⟩] ~Q )
6	1, 5	mpan2 519	. . 3 ⊢ (A ∈ N → [⟨(A ·N 1o), (A ·N 1o)⟩] ~Q = [⟨1o, 1o⟩] ~Q )
7		mulidpi 3808	. . . . 5 ⊢ (A ∈ N → (A ·N 1o) = A)
8	7, 7	jca 236	. . . 4 ⊢ (A ∈ N → ((A ·N 1o) = A ∧ (A ·N 1o) = A))
9		opeq12 1878	. . . 4 ⊢ (((A ·N 1o) = A ∧ (A ·N 1o) = A) → ⟨(A ·N 1o), (A ·N 1o)⟩ = ⟨A, A⟩)
10		eceq2 3215	. . . 4 ⊢ (⟨(A ·N 1o), (A ·N 1o)⟩ = ⟨A, A⟩ → [⟨(A ·N 1o), (A ·N 1o)⟩] ~Q = [⟨A, A⟩] ~Q )
11	8, 9, 10	3syl 21	. . 3 ⊢ (A ∈ N → [⟨(A ·N 1o), (A ·N 1o)⟩] ~Q = [⟨A, A⟩] ~Q )
12	6, 11	eqtr3d 1130	. 2 ⊢ (A ∈ N → [⟨1o, 1o⟩] ~Q = [⟨A, A⟩] ~Q )
13		df-1q 3837	. 2 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
14	12, 13	syl5eq 1136	1 ⊢ (A ∈ N → 1Q = [⟨A, A⟩] ~Q )

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  (class class class)co 3001  1_oc1o 3099  [cec 3198  Ncnpi 3766   ·_N cmi 3768   ~_Q ceq 3772  1_Qc1q 3774

This theorem is referenced by:  recmulpq 3864

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-ni 3794  df-mi 3796  df-enq 3831  df-1q 3837

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