zringfrac - Mathbox for Thierry Arnoux

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Theorem zringfrac 33525

Description: The field of fractions of the ring of integers is isomorphic to the field of the rational numbers. (Contributed by Thierry Arnoux, 4-May-2025.)

**Hypotheses**
Ref	Expression
zringfrac.1	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
zringfrac.2	⊢ ∼ = (ℤ_ring ~_RL (ℤ ∖ {0}))
zringfrac.3	⊢ 𝐹 = (𝑞 ∈ ℚ ↦ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )

**Assertion**
Ref	Expression
zringfrac	⊢ 𝐹 ∈ (𝑄 RingIso ( Frac ‘ℤ_ring))

Distinct variable groups: ∼ ,𝑞 𝐹,𝑞 𝑄,𝑞

Proof of Theorem zringfrac Dummy variables 𝑎 𝑏 𝑧 𝑝 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zringfrac.1	. . . . . 6 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
2	1	qdrng 27531	. . . . 5 ⊢ 𝑄 ∈ DivRing
3		drngring 20645	. . . . 5 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
4	2, 3	ax-mp 5	. . . 4 ⊢ 𝑄 ∈ Ring
5		zringidom 33522	. . . . 5 ⊢ ℤ_ring ∈ IDomn
6		id 22	. . . . . . . 8 ⊢ (ℤ_ring ∈ IDomn → ℤ_ring ∈ IDomn)
7	6	fracfld 33258	. . . . . . 7 ⊢ (ℤ_ring ∈ IDomn → ( Frac ‘ℤ_ring) ∈ Field)
8	7	fldcrngd 20651	. . . . . 6 ⊢ (ℤ_ring ∈ IDomn → ( Frac ‘ℤ_ring) ∈ CRing)
9	8	crngringd 20155	. . . . 5 ⊢ (ℤ_ring ∈ IDomn → ( Frac ‘ℤ_ring) ∈ Ring)
10	5, 9	ax-mp 5	. . . 4 ⊢ ( Frac ‘ℤ_ring) ∈ Ring
11	4, 10	pm3.2i 470	. . 3 ⊢ (𝑄 ∈ Ring ∧ ( Frac ‘ℤ_ring) ∈ Ring)
12		ringgrp 20147	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Grp)
13	4, 12	ax-mp 5	. . . . . 6 ⊢ 𝑄 ∈ Grp
14		ringgrp 20147	. . . . . . 7 ⊢ (( Frac ‘ℤ_ring) ∈ Ring → ( Frac ‘ℤ_ring) ∈ Grp)
15	10, 14	ax-mp 5	. . . . . 6 ⊢ ( Frac ‘ℤ_ring) ∈ Grp
16	13, 15	pm3.2i 470	. . . . 5 ⊢ (𝑄 ∈ Grp ∧ ( Frac ‘ℤ_ring) ∈ Grp)
17		zringfrac.3	. . . . . . 7 ⊢ 𝐹 = (𝑞 ∈ ℚ ↦ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )
18		qnumcl 16710	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
19		qdencl 16711	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
20	19	nnzd 12556	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℤ)
21	19	nnne0d 12236	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ≠ 0)
22	20, 21	eldifsnd 4751	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ (ℤ ∖ {0}))
23	18, 22	opelxpd 5677	. . . . . . . . 9 ⊢ (𝑞 ∈ ℚ → ⟨(numer‘𝑞), (denom‘𝑞)⟩ ∈ (ℤ × (ℤ ∖ {0})))
24		zringfrac.2	. . . . . . . . . . 11 ⊢ ∼ = (ℤ_ring ~_RL (ℤ ∖ {0}))
25	24	ovexi 7421	. . . . . . . . . 10 ⊢ ∼ ∈ V
26	25	ecelqsi 8743	. . . . . . . . 9 ⊢ (⟨(numer‘𝑞), (denom‘𝑞)⟩ ∈ (ℤ × (ℤ ∖ {0})) → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ ((ℤ × (ℤ ∖ {0})) / ∼ ))
27	23, 26	syl 17	. . . . . . . 8 ⊢ (𝑞 ∈ ℚ → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ ((ℤ × (ℤ ∖ {0})) / ∼ ))
28		zringbas 21363	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤ_ring)
29		zring0 21368	. . . . . . . . . 10 ⊢ 0 = (0_g‘ℤ_ring)
30		zringmulr 21367	. . . . . . . . . 10 ⊢ · = (._r‘ℤ_ring)
31		eqid 2729	. . . . . . . . . 10 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
32		eqid 2729	. . . . . . . . . 10 ⊢ (ℤ × (ℤ ∖ {0})) = (ℤ × (ℤ ∖ {0}))
33		fracval 33254	. . . . . . . . . . 11 ⊢ ( Frac ‘ℤ_ring) = (ℤ_ring RLocal (RLReg‘ℤ_ring))
34	6	idomdomd 20635	. . . . . . . . . . . . . . 15 ⊢ (ℤ_ring ∈ IDomn → ℤ_ring ∈ Domn)
35	5, 34	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ℤ_ring ∈ Domn
36		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (RLReg‘ℤ_ring) = (RLReg‘ℤ_ring)
37	28, 36, 29	isdomn6 20623	. . . . . . . . . . . . . 14 ⊢ (ℤ_ring ∈ Domn ↔ (ℤ_ring ∈ NzRing ∧ (ℤ ∖ {0}) = (RLReg‘ℤ_ring)))
38	35, 37	mpbi 230	. . . . . . . . . . . . 13 ⊢ (ℤ_ring ∈ NzRing ∧ (ℤ ∖ {0}) = (RLReg‘ℤ_ring))
39	38	simpri 485	. . . . . . . . . . . 12 ⊢ (ℤ ∖ {0}) = (RLReg‘ℤ_ring)
40	39	oveq2i 7398	. . . . . . . . . . 11 ⊢ (ℤ_ring RLocal (ℤ ∖ {0})) = (ℤ_ring RLocal (RLReg‘ℤ_ring))
41	33, 40	eqtr4i 2755	. . . . . . . . . 10 ⊢ ( Frac ‘ℤ_ring) = (ℤ_ring RLocal (ℤ ∖ {0}))
42	5	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℤ_ring ∈ IDomn)
43		difssd 4100	. . . . . . . . . 10 ⊢ (⊤ → (ℤ ∖ {0}) ⊆ ℤ)
44	28, 29, 30, 31, 32, 41, 24, 42, 43	rlocbas 33218	. . . . . . . . 9 ⊢ (⊤ → ((ℤ × (ℤ ∖ {0})) / ∼ ) = (Base‘( Frac ‘ℤ_ring)))
45	44	mptru 1547	. . . . . . . 8 ⊢ ((ℤ × (ℤ ∖ {0})) / ∼ ) = (Base‘( Frac ‘ℤ_ring))
46	27, 45	eleqtrdi 2838	. . . . . . 7 ⊢ (𝑞 ∈ ℚ → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ (Base‘( Frac ‘ℤ_ring)))
47	17, 46	fmpti 7084	. . . . . 6 ⊢ 𝐹:ℚ⟶(Base‘( Frac ‘ℤ_ring))
48		ecexg 8675	. . . . . . . . . . . 12 ⊢ ( ∼ ∈ V → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ V)
49	25, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ V
50	17	fvmpt2 6979	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℚ ∧ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ V) → (𝐹‘𝑞) = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )
51	49, 50	mpan2 691	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℚ → (𝐹‘𝑞) = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )
52	51	adantr 480	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝐹‘𝑞) = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )
53		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑝 → (numer‘𝑞) = (numer‘𝑝))
54		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑝 → (denom‘𝑞) = (denom‘𝑝))
55	53, 54	opeq12d 4845	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑝 → ⟨(numer‘𝑞), (denom‘𝑞)⟩ = ⟨(numer‘𝑝), (denom‘𝑝)⟩)
56	55	eceq1d 8711	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑝 → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ )
57	56, 17, 27	fvmpt3 6972	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℚ → (𝐹‘𝑝) = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ )
58	57	adantl 481	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝐹‘𝑝) = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ )
59	52, 58	oveq12d 7405	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝐹‘𝑞)(+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))(𝐹‘𝑝)) = ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ))
60	41	fveq2i 6861	. . . . . . . . . 10 ⊢ (+_g‘( Frac ‘ℤ_ring)) = (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))
61	60	oveqi 7400	. . . . . . . . 9 ⊢ ((𝐹‘𝑞)(+_g‘( Frac ‘ℤ_ring))(𝐹‘𝑝)) = ((𝐹‘𝑞)(+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))(𝐹‘𝑝))
62	61	a1i 11	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝐹‘𝑞)(+_g‘( Frac ‘ℤ_ring))(𝐹‘𝑝)) = ((𝐹‘𝑞)(+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))(𝐹‘𝑝)))
63		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑢 → (numer‘𝑞) = (numer‘𝑢))
64		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑢 → (denom‘𝑞) = (denom‘𝑢))
65	63, 64	opeq12d 4845	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑢 → ⟨(numer‘𝑞), (denom‘𝑞)⟩ = ⟨(numer‘𝑢), (denom‘𝑢)⟩)
66	65	eceq1d 8711	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑢 → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑢), (denom‘𝑢)⟩] ∼ )
67	66	cbvmptv 5211	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℚ ↦ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ) = (𝑢 ∈ ℚ ↦ [⟨(numer‘𝑢), (denom‘𝑢)⟩] ∼ )
68	17, 67	eqtri 2752	. . . . . . . . 9 ⊢ 𝐹 = (𝑢 ∈ ℚ ↦ [⟨(numer‘𝑢), (denom‘𝑢)⟩] ∼ )
69		zring1 21369	. . . . . . . . . . . . 13 ⊢ 1 = (1_r‘ℤ_ring)
70	5	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ℤ_ring ∈ IDomn)
71	70	idomcringd 20636	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ℤ_ring ∈ CRing)
72	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ℤ_ring ∈ Domn)
73		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (mulGrp‘ℤ_ring) = (mulGrp‘ℤ_ring)
74	28, 29, 73	isdomn3 20624	. . . . . . . . . . . . . . 15 ⊢ (ℤ_ring ∈ Domn ↔ (ℤ_ring ∈ Ring ∧ (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring))))
75	72, 74	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (ℤ_ring ∈ Ring ∧ (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring))))
76	75	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring)))
77	28, 29, 69, 30, 31, 32, 24, 71, 76	erler 33216	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ∼ Er (ℤ × (ℤ ∖ {0})))
78		qcn 12922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ ℚ → 𝑞 ∈ ℂ)
79	78	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → 𝑞 ∈ ℂ)
80		qcn 12922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ℚ → 𝑝 ∈ ℂ)
81	80	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → 𝑝 ∈ ℂ)
82	79, 81	addcld 11193	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 + 𝑝) ∈ ℂ)
83		qaddcl 12924	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 + 𝑝) ∈ ℚ)
84		qdencl 16711	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 + 𝑝) ∈ ℚ → (denom‘(𝑞 + 𝑝)) ∈ ℕ)
85	83, 84	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 + 𝑝)) ∈ ℕ)
86	85	nncnd 12202	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 + 𝑝)) ∈ ℂ)
87	19	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑞) ∈ ℕ)
88	87	nncnd 12202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑞) ∈ ℂ)
89		qdencl 16711	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ ℚ → (denom‘𝑝) ∈ ℕ)
90	89	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑝) ∈ ℕ)
91	90	nncnd 12202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑝) ∈ ℂ)
92	88, 91	mulcld 11194	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑞) · (denom‘𝑝)) ∈ ℂ)
93	82, 86, 92	mul32d 11384	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 + 𝑝) · (denom‘(𝑞 + 𝑝))) · ((denom‘𝑞) · (denom‘𝑝))) = (((𝑞 + 𝑝) · ((denom‘𝑞) · (denom‘𝑝))) · (denom‘(𝑞 + 𝑝))))
94		qmuldeneqnum 16717	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 + 𝑝) ∈ ℚ → ((𝑞 + 𝑝) · (denom‘(𝑞 + 𝑝))) = (numer‘(𝑞 + 𝑝)))
95	83, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 + 𝑝) · (denom‘(𝑞 + 𝑝))) = (numer‘(𝑞 + 𝑝)))
96	95	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 + 𝑝) · (denom‘(𝑞 + 𝑝))) · ((denom‘𝑞) · (denom‘𝑝))) = ((numer‘(𝑞 + 𝑝)) · ((denom‘𝑞) · (denom‘𝑝))))
97	79, 88, 91	mulassd 11197	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 · (denom‘𝑞)) · (denom‘𝑝)) = (𝑞 · ((denom‘𝑞) · (denom‘𝑝))))
98		qmuldeneqnum 16717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ ℚ → (𝑞 · (denom‘𝑞)) = (numer‘𝑞))
99	98	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 · (denom‘𝑞)) = (numer‘𝑞))
100	99	oveq1d 7402	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 · (denom‘𝑞)) · (denom‘𝑝)) = ((numer‘𝑞) · (denom‘𝑝)))
101	97, 100	eqtr3d 2766	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 · ((denom‘𝑞) · (denom‘𝑝))) = ((numer‘𝑞) · (denom‘𝑝)))
102	81, 91, 88	mulassd 11197	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑝 · (denom‘𝑝)) · (denom‘𝑞)) = (𝑝 · ((denom‘𝑝) · (denom‘𝑞))))
103		qmuldeneqnum 16717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ ℚ → (𝑝 · (denom‘𝑝)) = (numer‘𝑝))
104	103	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑝 · (denom‘𝑝)) = (numer‘𝑝))
105	104	oveq1d 7402	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑝 · (denom‘𝑝)) · (denom‘𝑞)) = ((numer‘𝑝) · (denom‘𝑞)))
106	91, 88	mulcomd 11195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑝) · (denom‘𝑞)) = ((denom‘𝑞) · (denom‘𝑝)))
107	106	oveq2d 7403	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑝 · ((denom‘𝑝) · (denom‘𝑞))) = (𝑝 · ((denom‘𝑞) · (denom‘𝑝))))
108	102, 105, 107	3eqtr3rd 2773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑝 · ((denom‘𝑞) · (denom‘𝑝))) = ((numer‘𝑝) · (denom‘𝑞)))
109	101, 108	oveq12d 7405	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 · ((denom‘𝑞) · (denom‘𝑝))) + (𝑝 · ((denom‘𝑞) · (denom‘𝑝)))) = (((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))))
110	79, 92, 81, 109	joinlmuladdmuld 11201	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 + 𝑝) · ((denom‘𝑞) · (denom‘𝑝))) = (((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))))
111	110	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 + 𝑝) · ((denom‘𝑞) · (denom‘𝑝))) · (denom‘(𝑞 + 𝑝))) = ((((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))) · (denom‘(𝑞 + 𝑝))))
112	93, 96, 111	3eqtr3d 2772	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((numer‘(𝑞 + 𝑝)) · ((denom‘𝑞) · (denom‘𝑝))) = ((((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))) · (denom‘(𝑞 + 𝑝))))
113	39	oveq2i 7398	. . . . . . . . . . . . . . 15 ⊢ (ℤ_ring ~_RL (ℤ ∖ {0})) = (ℤ_ring ~_RL (RLReg‘ℤ_ring))
114	24, 113	eqtri 2752	. . . . . . . . . . . . . 14 ⊢ ∼ = (ℤ_ring ~_RL (RLReg‘ℤ_ring))
115		qnumcl 16710	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 + 𝑝) ∈ ℚ → (numer‘(𝑞 + 𝑝)) ∈ ℤ)
116	83, 115	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (numer‘(𝑞 + 𝑝)) ∈ ℤ)
117	18	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (numer‘𝑞) ∈ ℤ)
118	89	nnzd 12556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ℚ → (denom‘𝑝) ∈ ℤ)
119	118	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑝) ∈ ℤ)
120	117, 119	zmulcld 12644	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((numer‘𝑞) · (denom‘𝑝)) ∈ ℤ)
121		qnumcl 16710	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ℚ → (numer‘𝑝) ∈ ℤ)
122	121	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (numer‘𝑝) ∈ ℤ)
123	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑞) ∈ ℤ)
124	122, 123	zmulcld 12644	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((numer‘𝑝) · (denom‘𝑞)) ∈ ℤ)
125	120, 124	zaddcld 12642	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))) ∈ ℤ)
126	85	nnzd 12556	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 + 𝑝)) ∈ ℤ)
127	85	nnne0d 12236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 + 𝑝)) ≠ 0)
128	126, 127	eldifsnd 4751	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 + 𝑝)) ∈ (ℤ ∖ {0}))
129	128, 39	eleqtrdi 2838	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 + 𝑝)) ∈ (RLReg‘ℤ_ring))
130	123, 119	zmulcld 12644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑞) · (denom‘𝑝)) ∈ ℤ)
131	87, 90	nnmulcld 12239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑞) · (denom‘𝑝)) ∈ ℕ)
132	131	nnne0d 12236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑞) · (denom‘𝑝)) ≠ 0)
133	130, 132	eldifsnd 4751	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑞) · (denom‘𝑝)) ∈ (ℤ ∖ {0}))
134	133, 39	eleqtrdi 2838	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((denom‘𝑞) · (denom‘𝑝)) ∈ (RLReg‘ℤ_ring))
135	28, 30, 114, 71, 116, 125, 129, 134	fracerl 33256	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩ ∼ ⟨(((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))), ((denom‘𝑞) · (denom‘𝑝))⟩ ↔ ((numer‘(𝑞 + 𝑝)) · ((denom‘𝑞) · (denom‘𝑝))) = ((((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))) · (denom‘(𝑞 + 𝑝)))))
136	112, 135	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩ ∼ ⟨(((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))), ((denom‘𝑞) · (denom‘𝑝))⟩)
137	77, 136	erthi 8727	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → [⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩] ∼ = [⟨(((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))), ((denom‘𝑞) · (denom‘𝑝))⟩] ∼ )
138	137	adantr 480	. . . . . . . . . 10 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → [⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩] ∼ = [⟨(((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))), ((denom‘𝑞) · (denom‘𝑝))⟩] ∼ )
139		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑞 + 𝑝) → (numer‘𝑢) = (numer‘(𝑞 + 𝑝)))
140		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑞 + 𝑝) → (denom‘𝑢) = (denom‘(𝑞 + 𝑝)))
141	139, 140	opeq12d 4845	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑞 + 𝑝) → ⟨(numer‘𝑢), (denom‘𝑢)⟩ = ⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩)
142	141	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → ⟨(numer‘𝑢), (denom‘𝑢)⟩ = ⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩)
143	142	eceq1d 8711	. . . . . . . . . 10 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → [⟨(numer‘𝑢), (denom‘𝑢)⟩] ∼ = [⟨(numer‘(𝑞 + 𝑝)), (denom‘(𝑞 + 𝑝))⟩] ∼ )
144		zringplusg 21364	. . . . . . . . . . 11 ⊢ + = (+_g‘ℤ_ring)
145		eqid 2729	. . . . . . . . . . 11 ⊢ (ℤ_ring RLocal (ℤ ∖ {0})) = (ℤ_ring RLocal (ℤ ∖ {0}))
146		zringcrng 21358	. . . . . . . . . . . 12 ⊢ ℤ_ring ∈ CRing
147	146	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → ℤ_ring ∈ CRing)
148	35, 74	mpbi 230	. . . . . . . . . . . . 13 ⊢ (ℤ_ring ∈ Ring ∧ (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring)))
149	148	simpri 485	. . . . . . . . . . . 12 ⊢ (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring))
150	149	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring)))
151	117	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → (numer‘𝑞) ∈ ℤ)
152	122	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → (numer‘𝑝) ∈ ℤ)
153	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑞) ∈ (ℤ ∖ {0}))
154	153	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → (denom‘𝑞) ∈ (ℤ ∖ {0}))
155	89	nnne0d 12236	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℚ → (denom‘𝑝) ≠ 0)
156	118, 155	eldifsnd 4751	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℚ → (denom‘𝑝) ∈ (ℤ ∖ {0}))
157	156	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑝) ∈ (ℤ ∖ {0}))
158	157	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → (denom‘𝑝) ∈ (ℤ ∖ {0}))
159		eqid 2729	. . . . . . . . . . 11 ⊢ (+_g‘(ℤ_ring RLocal (ℤ ∖ {0}))) = (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))
160	28, 30, 144, 145, 24, 147, 150, 151, 152, 154, 158, 159	rlocaddval 33219	. . . . . . . . . 10 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) = [⟨(((numer‘𝑞) · (denom‘𝑝)) + ((numer‘𝑝) · (denom‘𝑞))), ((denom‘𝑞) · (denom‘𝑝))⟩] ∼ )
161	138, 143, 160	3eqtr4d 2774	. . . . . . . . 9 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ 𝑢 = (𝑞 + 𝑝)) → [⟨(numer‘𝑢), (denom‘𝑢)⟩] ∼ = ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ))
162		ovexd 7422	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) ∈ V)
163	68, 161, 83, 162	fvmptd2 6976	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝐹‘(𝑞 + 𝑝)) = ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (+_g‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ))
164	59, 62, 163	3eqtr4rd 2775	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝐹‘(𝑞 + 𝑝)) = ((𝐹‘𝑞)(+_g‘( Frac ‘ℤ_ring))(𝐹‘𝑝)))
165	164	rgen2 3177	. . . . . 6 ⊢ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ (𝐹‘(𝑞 + 𝑝)) = ((𝐹‘𝑞)(+_g‘( Frac ‘ℤ_ring))(𝐹‘𝑝))
166	47, 165	pm3.2i 470	. . . . 5 ⊢ (𝐹:ℚ⟶(Base‘( Frac ‘ℤ_ring)) ∧ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ (𝐹‘(𝑞 + 𝑝)) = ((𝐹‘𝑞)(+_g‘( Frac ‘ℤ_ring))(𝐹‘𝑝)))
167	1	qrngbas 27530	. . . . . 6 ⊢ ℚ = (Base‘𝑄)
168		eqid 2729	. . . . . 6 ⊢ (Base‘( Frac ‘ℤ_ring)) = (Base‘( Frac ‘ℤ_ring))
169		qex 12920	. . . . . . 7 ⊢ ℚ ∈ V
170		cnfldadd 21270	. . . . . . . 8 ⊢ + = (+_g‘ℂ_fld)
171	1, 170	ressplusg 17254	. . . . . . 7 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
172	169, 171	ax-mp 5	. . . . . 6 ⊢ + = (+_g‘𝑄)
173		eqid 2729	. . . . . 6 ⊢ (+_g‘( Frac ‘ℤ_ring)) = (+_g‘( Frac ‘ℤ_ring))
174	167, 168, 172, 173	isghm 19147	. . . . 5 ⊢ (𝐹 ∈ (𝑄 GrpHom ( Frac ‘ℤ_ring)) ↔ ((𝑄 ∈ Grp ∧ ( Frac ‘ℤ_ring) ∈ Grp) ∧ (𝐹:ℚ⟶(Base‘( Frac ‘ℤ_ring)) ∧ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ (𝐹‘(𝑞 + 𝑝)) = ((𝐹‘𝑞)(+_g‘( Frac ‘ℤ_ring))(𝐹‘𝑝)))))
175	16, 166, 174	mpbir2an 711	. . . 4 ⊢ 𝐹 ∈ (𝑄 GrpHom ( Frac ‘ℤ_ring))
176		eqid 2729	. . . . . . . 8 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
177	176	ringmgp 20148	. . . . . . 7 ⊢ (𝑄 ∈ Ring → (mulGrp‘𝑄) ∈ Mnd)
178	4, 177	ax-mp 5	. . . . . 6 ⊢ (mulGrp‘𝑄) ∈ Mnd
179		eqid 2729	. . . . . . . 8 ⊢ (mulGrp‘( Frac ‘ℤ_ring)) = (mulGrp‘( Frac ‘ℤ_ring))
180	179	ringmgp 20148	. . . . . . 7 ⊢ (( Frac ‘ℤ_ring) ∈ Ring → (mulGrp‘( Frac ‘ℤ_ring)) ∈ Mnd)
181	10, 180	ax-mp 5	. . . . . 6 ⊢ (mulGrp‘( Frac ‘ℤ_ring)) ∈ Mnd
182	178, 181	pm3.2i 470	. . . . 5 ⊢ ((mulGrp‘𝑄) ∈ Mnd ∧ (mulGrp‘( Frac ‘ℤ_ring)) ∈ Mnd)
183		eqid 2729	. . . . . . . . . 10 ⊢ (._r‘(ℤ_ring RLocal (ℤ ∖ {0}))) = (._r‘(ℤ_ring RLocal (ℤ ∖ {0})))
184	28, 30, 144, 145, 24, 71, 76, 117, 122, 153, 157, 183	rlocmulval 33220	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (._r‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) = [⟨((numer‘𝑞) · (numer‘𝑝)), ((denom‘𝑞) · (denom‘𝑝))⟩] ∼ )
185	79, 81	mulcld 11194	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 · 𝑝) ∈ ℂ)
186		qmulcl 12926	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 · 𝑝) ∈ ℚ)
187		qdencl 16711	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 · 𝑝) ∈ ℚ → (denom‘(𝑞 · 𝑝)) ∈ ℕ)
188	186, 187	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 · 𝑝)) ∈ ℕ)
189	188	nncnd 12202	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 · 𝑝)) ∈ ℂ)
190	185, 189, 92	mul32d 11384	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 · 𝑝) · (denom‘(𝑞 · 𝑝))) · ((denom‘𝑞) · (denom‘𝑝))) = (((𝑞 · 𝑝) · ((denom‘𝑞) · (denom‘𝑝))) · (denom‘(𝑞 · 𝑝))))
191	79, 81, 88, 91	mul4d 11386	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 · 𝑝) · ((denom‘𝑞) · (denom‘𝑝))) = ((𝑞 · (denom‘𝑞)) · (𝑝 · (denom‘𝑝))))
192	191	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 · 𝑝) · ((denom‘𝑞) · (denom‘𝑝))) · (denom‘(𝑞 · 𝑝))) = (((𝑞 · (denom‘𝑞)) · (𝑝 · (denom‘𝑝))) · (denom‘(𝑞 · 𝑝))))
193	190, 192	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 · 𝑝) · (denom‘(𝑞 · 𝑝))) · ((denom‘𝑞) · (denom‘𝑝))) = (((𝑞 · (denom‘𝑞)) · (𝑝 · (denom‘𝑝))) · (denom‘(𝑞 · 𝑝))))
194		qmuldeneqnum 16717	. . . . . . . . . . . . . 14 ⊢ ((𝑞 · 𝑝) ∈ ℚ → ((𝑞 · 𝑝) · (denom‘(𝑞 · 𝑝))) = (numer‘(𝑞 · 𝑝)))
195	186, 194	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 · 𝑝) · (denom‘(𝑞 · 𝑝))) = (numer‘(𝑞 · 𝑝)))
196	195	oveq1d 7402	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 · 𝑝) · (denom‘(𝑞 · 𝑝))) · ((denom‘𝑞) · (denom‘𝑝))) = ((numer‘(𝑞 · 𝑝)) · ((denom‘𝑞) · (denom‘𝑝))))
197	99, 104	oveq12d 7405	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝑞 · (denom‘𝑞)) · (𝑝 · (denom‘𝑝))) = ((numer‘𝑞) · (numer‘𝑝)))
198	197	oveq1d 7402	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((𝑞 · (denom‘𝑞)) · (𝑝 · (denom‘𝑝))) · (denom‘(𝑞 · 𝑝))) = (((numer‘𝑞) · (numer‘𝑝)) · (denom‘(𝑞 · 𝑝))))
199	193, 196, 198	3eqtr3rd 2773	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((numer‘𝑞) · (numer‘𝑝)) · (denom‘(𝑞 · 𝑝))) = ((numer‘(𝑞 · 𝑝)) · ((denom‘𝑞) · (denom‘𝑝))))
200	117, 122	zmulcld 12644	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((numer‘𝑞) · (numer‘𝑝)) ∈ ℤ)
201		qnumcl 16710	. . . . . . . . . . . . 13 ⊢ ((𝑞 · 𝑝) ∈ ℚ → (numer‘(𝑞 · 𝑝)) ∈ ℤ)
202	186, 201	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (numer‘(𝑞 · 𝑝)) ∈ ℤ)
203	188	nnzd 12556	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 · 𝑝)) ∈ ℤ)
204	188	nnne0d 12236	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 · 𝑝)) ≠ 0)
205	203, 204	eldifsnd 4751	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 · 𝑝)) ∈ (ℤ ∖ {0}))
206	205, 39	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘(𝑞 · 𝑝)) ∈ (RLReg‘ℤ_ring))
207	28, 30, 114, 71, 200, 202, 134, 206	fracerl 33256	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (⟨((numer‘𝑞) · (numer‘𝑝)), ((denom‘𝑞) · (denom‘𝑝))⟩ ∼ ⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩ ↔ (((numer‘𝑞) · (numer‘𝑝)) · (denom‘(𝑞 · 𝑝))) = ((numer‘(𝑞 · 𝑝)) · ((denom‘𝑞) · (denom‘𝑝)))))
208	199, 207	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ⟨((numer‘𝑞) · (numer‘𝑝)), ((denom‘𝑞) · (denom‘𝑝))⟩ ∼ ⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩)
209	77, 208	erthi 8727	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → [⟨((numer‘𝑞) · (numer‘𝑝)), ((denom‘𝑞) · (denom‘𝑝))⟩] ∼ = [⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩] ∼ )
210	184, 209	eqtrd 2764	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (._r‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) = [⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩] ∼ )
211	41	fveq2i 6861	. . . . . . . . . 10 ⊢ (._r‘( Frac ‘ℤ_ring)) = (._r‘(ℤ_ring RLocal (ℤ ∖ {0})))
212	211	a1i 11	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (._r‘( Frac ‘ℤ_ring)) = (._r‘(ℤ_ring RLocal (ℤ ∖ {0}))))
213	212, 52, 58	oveq123d 7408	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((𝐹‘𝑞)(._r‘( Frac ‘ℤ_ring))(𝐹‘𝑝)) = ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ (._r‘(ℤ_ring RLocal (ℤ ∖ {0})))[⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ))
214		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑞 · 𝑝) → (numer‘𝑢) = (numer‘(𝑞 · 𝑝)))
215		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑞 · 𝑝) → (denom‘𝑢) = (denom‘(𝑞 · 𝑝)))
216	214, 215	opeq12d 4845	. . . . . . . . . 10 ⊢ (𝑢 = (𝑞 · 𝑝) → ⟨(numer‘𝑢), (denom‘𝑢)⟩ = ⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩)
217	216	eceq1d 8711	. . . . . . . . 9 ⊢ (𝑢 = (𝑞 · 𝑝) → [⟨(numer‘𝑢), (denom‘𝑢)⟩] ∼ = [⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩] ∼ )
218		ecexg 8675	. . . . . . . . . 10 ⊢ ( ∼ ∈ V → [⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩] ∼ ∈ V)
219	25, 218	mp1i 13	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → [⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩] ∼ ∈ V)
220	68, 217, 186, 219	fvmptd3 6991	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝐹‘(𝑞 · 𝑝)) = [⟨(numer‘(𝑞 · 𝑝)), (denom‘(𝑞 · 𝑝))⟩] ∼ )
221	210, 213, 220	3eqtr4rd 2775	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝐹‘(𝑞 · 𝑝)) = ((𝐹‘𝑞)(._r‘( Frac ‘ℤ_ring))(𝐹‘𝑝)))
222	221	rgen2 3177	. . . . . 6 ⊢ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ (𝐹‘(𝑞 · 𝑝)) = ((𝐹‘𝑞)(._r‘( Frac ‘ℤ_ring))(𝐹‘𝑝))
223		zssq 12915	. . . . . . . 8 ⊢ ℤ ⊆ ℚ
224		1z 12563	. . . . . . . 8 ⊢ 1 ∈ ℤ
225	223, 224	sselii 3943	. . . . . . 7 ⊢ 1 ∈ ℚ
226		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑞 = 1 → (numer‘𝑞) = (numer‘1))
227		1zzd 12564	. . . . . . . . . . . . 13 ⊢ (ℤ_ring ∈ IDomn → 1 ∈ ℤ)
228	227	znumd 32737	. . . . . . . . . . . 12 ⊢ (ℤ_ring ∈ IDomn → (numer‘1) = 1)
229	5, 228	ax-mp 5	. . . . . . . . . . 11 ⊢ (numer‘1) = 1
230	226, 229	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑞 = 1 → (numer‘𝑞) = 1)
231		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑞 = 1 → (denom‘𝑞) = (denom‘1))
232	227	zdend 32738	. . . . . . . . . . . 12 ⊢ (ℤ_ring ∈ IDomn → (denom‘1) = 1)
233	5, 232	ax-mp 5	. . . . . . . . . . 11 ⊢ (denom‘1) = 1
234	231, 233	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑞 = 1 → (denom‘𝑞) = 1)
235	230, 234	opeq12d 4845	. . . . . . . . 9 ⊢ (𝑞 = 1 → ⟨(numer‘𝑞), (denom‘𝑞)⟩ = ⟨1, 1⟩)
236	235	eceq1d 8711	. . . . . . . 8 ⊢ (𝑞 = 1 → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨1, 1⟩] ∼ )
237	236, 17, 49	fvmpt3i 6973	. . . . . . 7 ⊢ (1 ∈ ℚ → (𝐹‘1) = [⟨1, 1⟩] ∼ )
238	225, 237	ax-mp 5	. . . . . 6 ⊢ (𝐹‘1) = [⟨1, 1⟩] ∼
239	47, 222, 238	3pm3.2i 1340	. . . . 5 ⊢ (𝐹:ℚ⟶(Base‘( Frac ‘ℤ_ring)) ∧ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ (𝐹‘(𝑞 · 𝑝)) = ((𝐹‘𝑞)(._r‘( Frac ‘ℤ_ring))(𝐹‘𝑝)) ∧ (𝐹‘1) = [⟨1, 1⟩] ∼ )
240	176, 167	mgpbas 20054	. . . . . 6 ⊢ ℚ = (Base‘(mulGrp‘𝑄))
241	179, 168	mgpbas 20054	. . . . . 6 ⊢ (Base‘( Frac ‘ℤ_ring)) = (Base‘(mulGrp‘( Frac ‘ℤ_ring)))
242		cnfldmul 21272	. . . . . . . . 9 ⊢ · = (._r‘ℂ_fld)
243	1, 242	ressmulr 17270	. . . . . . . 8 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
244	169, 243	ax-mp 5	. . . . . . 7 ⊢ · = (._r‘𝑄)
245	176, 244	mgpplusg 20053	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘𝑄))
246		eqid 2729	. . . . . . 7 ⊢ (._r‘( Frac ‘ℤ_ring)) = (._r‘( Frac ‘ℤ_ring))
247	179, 246	mgpplusg 20053	. . . . . 6 ⊢ (._r‘( Frac ‘ℤ_ring)) = (+_g‘(mulGrp‘( Frac ‘ℤ_ring)))
248	1	qrng1 27533	. . . . . . 7 ⊢ 1 = (1_r‘𝑄)
249	176, 248	ringidval 20092	. . . . . 6 ⊢ 1 = (0_g‘(mulGrp‘𝑄))
250	146	a1i 11	. . . . . . . . 9 ⊢ (ℤ_ring ∈ IDomn → ℤ_ring ∈ CRing)
251	149	a1i 11	. . . . . . . . 9 ⊢ (ℤ_ring ∈ IDomn → (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring)))
252		eqid 2729	. . . . . . . . 9 ⊢ [⟨1, 1⟩] ∼ = [⟨1, 1⟩] ∼
253	29, 69, 41, 24, 250, 251, 252	rloc1r 33223	. . . . . . . 8 ⊢ (ℤ_ring ∈ IDomn → [⟨1, 1⟩] ∼ = (1_r‘( Frac ‘ℤ_ring)))
254	5, 253	ax-mp 5	. . . . . . 7 ⊢ [⟨1, 1⟩] ∼ = (1_r‘( Frac ‘ℤ_ring))
255	179, 254	ringidval 20092	. . . . . 6 ⊢ [⟨1, 1⟩] ∼ = (0_g‘(mulGrp‘( Frac ‘ℤ_ring)))
256	240, 241, 245, 247, 249, 255	ismhm 18712	. . . . 5 ⊢ (𝐹 ∈ ((mulGrp‘𝑄) MndHom (mulGrp‘( Frac ‘ℤ_ring))) ↔ (((mulGrp‘𝑄) ∈ Mnd ∧ (mulGrp‘( Frac ‘ℤ_ring)) ∈ Mnd) ∧ (𝐹:ℚ⟶(Base‘( Frac ‘ℤ_ring)) ∧ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ (𝐹‘(𝑞 · 𝑝)) = ((𝐹‘𝑞)(._r‘( Frac ‘ℤ_ring))(𝐹‘𝑝)) ∧ (𝐹‘1) = [⟨1, 1⟩] ∼ )))
257	182, 239, 256	mpbir2an 711	. . . 4 ⊢ 𝐹 ∈ ((mulGrp‘𝑄) MndHom (mulGrp‘( Frac ‘ℤ_ring)))
258	175, 257	pm3.2i 470	. . 3 ⊢ (𝐹 ∈ (𝑄 GrpHom ( Frac ‘ℤ_ring)) ∧ 𝐹 ∈ ((mulGrp‘𝑄) MndHom (mulGrp‘( Frac ‘ℤ_ring))))
259	176, 179	isrhm 20387	. . 3 ⊢ (𝐹 ∈ (𝑄 RingHom ( Frac ‘ℤ_ring)) ↔ ((𝑄 ∈ Ring ∧ ( Frac ‘ℤ_ring) ∈ Ring) ∧ (𝐹 ∈ (𝑄 GrpHom ( Frac ‘ℤ_ring)) ∧ 𝐹 ∈ ((mulGrp‘𝑄) MndHom (mulGrp‘( Frac ‘ℤ_ring))))))
260	11, 258, 259	mpbir2an 711	. 2 ⊢ 𝐹 ∈ (𝑄 RingHom ( Frac ‘ℤ_ring))
261	46	rgen 3046	. . . 4 ⊢ ∀𝑞 ∈ ℚ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ (Base‘( Frac ‘ℤ_ring))
262	117	zcnd 12639	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (numer‘𝑞) ∈ ℂ)
263	122	zcnd 12639	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (numer‘𝑝) ∈ ℂ)
264	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑞) ≠ 0)
265	155	adantl 481	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑝) ≠ 0)
266	262, 88, 263, 91, 264, 265	divmuleqd 12004	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (((numer‘𝑞) / (denom‘𝑞)) = ((numer‘𝑝) / (denom‘𝑝)) ↔ ((numer‘𝑞) · (denom‘𝑝)) = ((numer‘𝑝) · (denom‘𝑞))))
267	153, 39	eleqtrdi 2838	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑞) ∈ (RLReg‘ℤ_ring))
268	157, 39	eleqtrdi 2838	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (denom‘𝑝) ∈ (RLReg‘ℤ_ring))
269	28, 30, 114, 71, 117, 122, 267, 268	fracerl 33256	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (⟨(numer‘𝑞), (denom‘𝑞)⟩ ∼ ⟨(numer‘𝑝), (denom‘𝑝)⟩ ↔ ((numer‘𝑞) · (denom‘𝑝)) = ((numer‘𝑝) · (denom‘𝑞))))
270	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ⟨(numer‘𝑞), (denom‘𝑞)⟩ ∈ (ℤ × (ℤ ∖ {0})))
271	77, 270	erth 8725	. . . . . . . . 9 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (⟨(numer‘𝑞), (denom‘𝑞)⟩ ∼ ⟨(numer‘𝑝), (denom‘𝑝)⟩ ↔ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ))
272	266, 269, 271	3bitr2rd 308	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ↔ ((numer‘𝑞) / (denom‘𝑞)) = ((numer‘𝑝) / (denom‘𝑝))))
273	272	biimpa 476	. . . . . . 7 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) → ((numer‘𝑞) / (denom‘𝑞)) = ((numer‘𝑝) / (denom‘𝑝)))
274		qeqnumdivden 16716	. . . . . . . 8 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
275	274	ad2antrr 726	. . . . . . 7 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
276		qeqnumdivden 16716	. . . . . . . 8 ⊢ (𝑝 ∈ ℚ → 𝑝 = ((numer‘𝑝) / (denom‘𝑝)))
277	276	ad2antlr 727	. . . . . . 7 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) → 𝑝 = ((numer‘𝑝) / (denom‘𝑝)))
278	273, 275, 277	3eqtr4d 2774	. . . . . 6 ⊢ (((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) ∧ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ ) → 𝑞 = 𝑝)
279	278	ex 412	. . . . 5 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ → 𝑞 = 𝑝))
280	279	rgen2 3177	. . . 4 ⊢ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ → 𝑞 = 𝑝)
281	17, 56	f1mpt 7236	. . . 4 ⊢ (𝐹:ℚ–1-1→(Base‘( Frac ‘ℤ_ring)) ↔ (∀𝑞 ∈ ℚ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ (Base‘( Frac ‘ℤ_ring)) ∧ ∀𝑞 ∈ ℚ ∀𝑝 ∈ ℚ ([⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘𝑝), (denom‘𝑝)⟩] ∼ → 𝑞 = 𝑝)))
282	261, 280, 281	mpbir2an 711	. . 3 ⊢ 𝐹:ℚ–1-1→(Base‘( Frac ‘ℤ_ring))
283		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑞 = (𝑎 / 𝑏) → (numer‘𝑞) = (numer‘(𝑎 / 𝑏)))
284		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑞 = (𝑎 / 𝑏) → (denom‘𝑞) = (denom‘(𝑎 / 𝑏)))
285	283, 284	opeq12d 4845	. . . . . . . . 9 ⊢ (𝑞 = (𝑎 / 𝑏) → ⟨(numer‘𝑞), (denom‘𝑞)⟩ = ⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩)
286	285	eceq1d 8711	. . . . . . . 8 ⊢ (𝑞 = (𝑎 / 𝑏) → [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ = [⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩] ∼ )
287	286	eqeq2d 2740	. . . . . . 7 ⊢ (𝑞 = (𝑎 / 𝑏) → (𝑧 = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ↔ 𝑧 = [⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩] ∼ ))
288		simpllr 775	. . . . . . . . 9 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑎 ∈ ℤ)
289	223, 288	sselid 3944	. . . . . . . 8 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑎 ∈ ℚ)
290		simplr 768	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ (ℤ ∖ {0}))
291	290	eldifad 3926	. . . . . . . . 9 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ ℤ)
292	223, 291	sselid 3944	. . . . . . . 8 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ ℚ)
293		eldifsni 4754	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℤ ∖ {0}) → 𝑏 ≠ 0)
294	290, 293	syl 17	. . . . . . . 8 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ≠ 0)
295		qdivcl 12929	. . . . . . . 8 ⊢ ((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ ∧ 𝑏 ≠ 0) → (𝑎 / 𝑏) ∈ ℚ)
296	289, 292, 294, 295	syl3anc 1373	. . . . . . 7 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑎 / 𝑏) ∈ ℚ)
297		simpr 484	. . . . . . . 8 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑧 = [⟨𝑎, 𝑏⟩] ∼ )
298	146	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → ℤ_ring ∈ CRing)
299	149	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → (ℤ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℤ_ring)))
300	28, 29, 69, 30, 31, 32, 24, 298, 299	erler 33216	. . . . . . . . 9 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → ∼ Er (ℤ × (ℤ ∖ {0})))
301		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑎 ∈ ℤ)
302	301	zcnd 12639	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑎 ∈ ℂ)
303		eldifi 4094	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (ℤ ∖ {0}) → 𝑏 ∈ ℤ)
304	303	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑏 ∈ ℤ)
305	304	zcnd 12639	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑏 ∈ ℂ)
306	293	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑏 ≠ 0)
307	302, 305, 306	divcld 11958	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (𝑎 / 𝑏) ∈ ℂ)
308	223, 301	sselid 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑎 ∈ ℚ)
309	223, 304	sselid 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑏 ∈ ℚ)
310	308, 309, 306, 295	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (𝑎 / 𝑏) ∈ ℚ)
311		qdencl 16711	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 / 𝑏) ∈ ℚ → (denom‘(𝑎 / 𝑏)) ∈ ℕ)
312	310, 311	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (denom‘(𝑎 / 𝑏)) ∈ ℕ)
313	312	nncnd 12202	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (denom‘(𝑎 / 𝑏)) ∈ ℂ)
314	307, 313, 305	mul32d 11384	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (((𝑎 / 𝑏) · (denom‘(𝑎 / 𝑏))) · 𝑏) = (((𝑎 / 𝑏) · 𝑏) · (denom‘(𝑎 / 𝑏))))
315		qmuldeneqnum 16717	. . . . . . . . . . . . . 14 ⊢ ((𝑎 / 𝑏) ∈ ℚ → ((𝑎 / 𝑏) · (denom‘(𝑎 / 𝑏))) = (numer‘(𝑎 / 𝑏)))
316	310, 315	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → ((𝑎 / 𝑏) · (denom‘(𝑎 / 𝑏))) = (numer‘(𝑎 / 𝑏)))
317	316	oveq1d 7402	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (((𝑎 / 𝑏) · (denom‘(𝑎 / 𝑏))) · 𝑏) = ((numer‘(𝑎 / 𝑏)) · 𝑏))
318	302, 305, 306	divcan1d 11959	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → ((𝑎 / 𝑏) · 𝑏) = 𝑎)
319	318	oveq1d 7402	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (((𝑎 / 𝑏) · 𝑏) · (denom‘(𝑎 / 𝑏))) = (𝑎 · (denom‘(𝑎 / 𝑏))))
320	314, 317, 319	3eqtr3rd 2773	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (𝑎 · (denom‘(𝑎 / 𝑏))) = ((numer‘(𝑎 / 𝑏)) · 𝑏))
321	146	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → ℤ_ring ∈ CRing)
322		qnumcl 16710	. . . . . . . . . . . . 13 ⊢ ((𝑎 / 𝑏) ∈ ℚ → (numer‘(𝑎 / 𝑏)) ∈ ℤ)
323	310, 322	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (numer‘(𝑎 / 𝑏)) ∈ ℤ)
324		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑏 ∈ (ℤ ∖ {0}))
325	324, 39	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → 𝑏 ∈ (RLReg‘ℤ_ring))
326	312	nnzd 12556	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (denom‘(𝑎 / 𝑏)) ∈ ℤ)
327	312	nnne0d 12236	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (denom‘(𝑎 / 𝑏)) ≠ 0)
328	326, 327	eldifsnd 4751	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (denom‘(𝑎 / 𝑏)) ∈ (ℤ ∖ {0}))
329	328, 39	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (denom‘(𝑎 / 𝑏)) ∈ (RLReg‘ℤ_ring))
330	28, 30, 114, 321, 301, 323, 325, 329	fracerl 33256	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → (⟨𝑎, 𝑏⟩ ∼ ⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩ ↔ (𝑎 · (denom‘(𝑎 / 𝑏))) = ((numer‘(𝑎 / 𝑏)) · 𝑏)))
331	320, 330	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ (ℤ ∖ {0})) → ⟨𝑎, 𝑏⟩ ∼ ⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩)
332	331	ad4ant23 753	. . . . . . . . 9 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → ⟨𝑎, 𝑏⟩ ∼ ⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩)
333	300, 332	erthi 8727	. . . . . . . 8 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → [⟨𝑎, 𝑏⟩] ∼ = [⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩] ∼ )
334	297, 333	eqtrd 2764	. . . . . . 7 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑧 = [⟨(numer‘(𝑎 / 𝑏)), (denom‘(𝑎 / 𝑏))⟩] ∼ )
335	287, 296, 334	rspcedvdw 3591	. . . . . 6 ⊢ ((((𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) ∧ 𝑎 ∈ ℤ) ∧ 𝑏 ∈ (ℤ ∖ {0})) ∧ 𝑧 = [⟨𝑎, 𝑏⟩] ∼ ) → ∃𝑞 ∈ ℚ 𝑧 = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )
336	45	eleq2i 2820	. . . . . . . 8 ⊢ (𝑧 ∈ ((ℤ × (ℤ ∖ {0})) / ∼ ) ↔ 𝑧 ∈ (Base‘( Frac ‘ℤ_ring)))
337	336	biimpri 228	. . . . . . 7 ⊢ (𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) → 𝑧 ∈ ((ℤ × (ℤ ∖ {0})) / ∼ ))
338	337	elrlocbasi 33217	. . . . . 6 ⊢ (𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ (ℤ ∖ {0})𝑧 = [⟨𝑎, 𝑏⟩] ∼ )
339	335, 338	r19.29vva 3197	. . . . 5 ⊢ (𝑧 ∈ (Base‘( Frac ‘ℤ_ring)) → ∃𝑞 ∈ ℚ 𝑧 = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )
340	339	rgen 3046	. . . 4 ⊢ ∀𝑧 ∈ (Base‘( Frac ‘ℤ_ring))∃𝑞 ∈ ℚ 𝑧 = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼
341	17	fompt 7090	. . . 4 ⊢ (𝐹:ℚ–onto→(Base‘( Frac ‘ℤ_ring)) ↔ (∀𝑞 ∈ ℚ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ∈ (Base‘( Frac ‘ℤ_ring)) ∧ ∀𝑧 ∈ (Base‘( Frac ‘ℤ_ring))∃𝑞 ∈ ℚ 𝑧 = [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ ))
342	261, 340, 341	mpbir2an 711	. . 3 ⊢ 𝐹:ℚ–onto→(Base‘( Frac ‘ℤ_ring))
343		df-f1o 6518	. . 3 ⊢ (𝐹:ℚ–1-1-onto→(Base‘( Frac ‘ℤ_ring)) ↔ (𝐹:ℚ–1-1→(Base‘( Frac ‘ℤ_ring)) ∧ 𝐹:ℚ–onto→(Base‘( Frac ‘ℤ_ring))))
344	282, 342, 343	mpbir2an 711	. 2 ⊢ 𝐹:ℚ–1-1-onto→(Base‘( Frac ‘ℤ_ring))
345	167, 168	isrim 20401	. 2 ⊢ (𝐹 ∈ (𝑄 RingIso ( Frac ‘ℤ_ring)) ↔ (𝐹 ∈ (𝑄 RingHom ( Frac ‘ℤ_ring)) ∧ 𝐹:ℚ–1-1-onto→(Base‘( Frac ‘ℤ_ring))))
346	260, 344, 345	mpbir2an 711	1 ⊢ 𝐹 ∈ (𝑄 RingIso ( Frac ‘ℤ_ring))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ∖ cdif 3911 {csn 4589 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 ⟶wf 6507 –1-1→wf1 6508 –onto→wfo 6509 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 [cec 8669 / cqs 8670 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 / cdiv 11835 ℕcn 12186 ℤcz 12529 ℚcq 12907 numercnumer 16703 denomcdenom 16704 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 ._rcmulr 17221 Mndcmnd 18661 MndHom cmhm 18708 SubMndcsubmnd 18709 Grpcgrp 18865 -_gcsg 18867 GrpHom cghm 19144 mulGrpcmgp 20049 1_rcur 20090 Ringcrg 20142 CRingccrg 20143 RingHom crh 20378 RingIso crs 20379 NzRingcnzr 20421 RLRegcrlreg 20600 Domncdomn 20601 IDomncidom 20602 DivRingcdr 20638 ℂ_fldccnfld 21264 ℤ_ringczring 21356 ~_RL cerl 33204 RLocal crloc 33205 Frac cfrac 33252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-numer 16705 df-denom 16706 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-imas 17471 df-qus 17472 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-ghm 19145 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-rim 20382 df-nzr 20422 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-domn 20604 df-idom 20605 df-drng 20640 df-field 20641 df-cnfld 21265 df-zring 21357 df-erl 33206 df-rloc 33207 df-frac 33253

This theorem is referenced by: (None)

W3C validator