qqhghm - Mathbox for Thierry Arnoux

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Theorem qqhghm 34731

Description: The ℚHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.)

**Hypotheses**
Ref	Expression
qqhval2.0	⊢ 𝐵 = (Base‘𝑅)
qqhval2.1	⊢ / = (/_r‘𝑅)
qqhval2.2	⊢ 𝐿 = (ℤRHom‘𝑅)
qqhrhm.1	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)

**Assertion**
Ref	Expression
qqhghm	⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))

Proof of Theorem qqhghm Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qqhrhm.1	. . 3 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
2	1	qrngbas 28003	. 2 ⊢ ℚ = (Base‘𝑄)
3		qqhval2.0	. 2 ⊢ 𝐵 = (Base‘𝑅)
4		qex 13128	. . 3 ⊢ ℚ ∈ V
5		cnfldadd 21735	. . . 4 ⊢ + = (+_g‘ℂ_fld)
6	1, 5	ressplusg 17508	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
7	4, 6	ax-mp 5	. 2 ⊢ + = (+_g‘𝑄)
8		eqid 2763	. 2 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
9	1	qdrng 28004	. . 3 ⊢ 𝑄 ∈ DivRing
10		drnggrp 21041	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Grp)
11	9, 10	mp1i 14	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑄 ∈ Grp)
12		drnggrp 21041	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)
13	12	adantr 486	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑅 ∈ Grp)
14		qqhval2.1	. . 3 ⊢ / = (/_r‘𝑅)
15		qqhval2.2	. . 3 ⊢ 𝐿 = (ℤRHom‘𝑅)
16	3, 14, 15	qqhf 34729	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
17		drngring 21038	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
18	17	ad2antrr 739	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ Ring)
19	17	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑅 ∈ Ring)
20	15	zrhrhm 21872	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
21		zringbas 21813	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
22	21, 3	rhmf 20766	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑅) → 𝐿:ℤ⟶𝐵)
23	19, 20, 22	3syl 19	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝐿:ℤ⟶𝐵)
24	23	adantr 486	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿:ℤ⟶𝐵)
25		qnumcl 16960	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → (numer‘𝑥) ∈ ℤ)
26	25	ad2antrl 741	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℤ)
27		qdencl 16961	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → (denom‘𝑦) ∈ ℕ)
28	27	ad2antll 742	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℕ)
29	28	nnzd 12759	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℤ)
30	26, 29	zmulcld 12849	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ)
31	24, 30	ffvelcdmd 7117	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑥) · (denom‘𝑦))) ∈ 𝐵)
32		qnumcl 16960	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (numer‘𝑦) ∈ ℤ)
33	32	ad2antll 742	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℤ)
34		qdencl 16961	. . . . . . . 8 ⊢ (𝑥 ∈ ℚ → (denom‘𝑥) ∈ ℕ)
35	34	ad2antrl 741	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℕ)
36	35	nnzd 12759	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℤ)
37	33, 36	zmulcld 12849	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ)
38	24, 37	ffvelcdmd 7117	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑦) · (denom‘𝑥))) ∈ 𝐵)
39	18, 20	syl 18	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
40		zringmulr 21818	. . . . . . 7 ⊢ · = (._r‘ℤ_ring)
41		eqid 2763	. . . . . . 7 ⊢ (._r‘𝑅) = (._r‘𝑅)
42	21, 40, 41	rhmmul 20771	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑅) ∧ (denom‘𝑥) ∈ ℤ ∧ (denom‘𝑦) ∈ ℤ) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))))
43	39, 36, 29, 42	syl3anc 1399	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))))
44		simpl 488	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0))
45	35	nnne0d 12428	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ≠ 0)
46		eqid 2763	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
47	3, 15, 46	elzrhunit 34720	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ≠ 0)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
48	44, 36, 45, 47	syl12anc 850	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
49	28	nnne0d 12428	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ≠ 0)
50	3, 15, 46	elzrhunit 34720	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ≠ 0)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
51	44, 29, 49, 50	syl12anc 850	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
52		eqid 2763	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
53	52, 41	unitmulcl 20661	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅) ∧ (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
54	18, 48, 51, 53	syl3anc 1399	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
55	43, 54	eqeltrd 2864	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) ∈ (Unit‘𝑅))
56	3, 52, 8, 14	dvrdir 20693	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝐿‘((numer‘𝑥) · (denom‘𝑦))) ∈ 𝐵 ∧ (𝐿‘((numer‘𝑦) · (denom‘𝑥))) ∈ 𝐵 ∧ (𝐿‘((denom‘𝑥) · (denom‘𝑦))) ∈ (Unit‘𝑅))) → (((𝐿‘((numer‘𝑥) · (denom‘𝑦)))(+_g‘𝑅)(𝐿‘((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))) = (((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦))))(+_g‘𝑅)((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦))))))
57	18, 31, 38, 55, 56	syl13anc 1400	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((𝐿‘((numer‘𝑥) · (denom‘𝑦)))(+_g‘𝑅)(𝐿‘((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))) = (((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦))))(+_g‘𝑅)((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦))))))
58		qeqnumdivden 16966	. . . . . . . 8 ⊢ (𝑥 ∈ ℚ → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
59	58	ad2antrl 741	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
60		qeqnumdivden 16966	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
61	60	ad2antll 742	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
62	59, 61	oveq12d 7469	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) = (((numer‘𝑥) / (denom‘𝑥)) + ((numer‘𝑦) / (denom‘𝑦))))
63	26	zcnd 12844	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℂ)
64	36	zcnd 12844	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℂ)
65	33	zcnd 12844	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℂ)
66	29	zcnd 12844	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℂ)
67	63, 64, 65, 66, 45, 49	divadddivd 12177	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) / (denom‘𝑥)) + ((numer‘𝑦) / (denom‘𝑦))) = ((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) / ((denom‘𝑥) · (denom‘𝑦))))
68	62, 67	eqtrd 2798	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) = ((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) / ((denom‘𝑥) · (denom‘𝑦))))
69	68	fveq2d 6920	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 + 𝑦)) = ((ℚHom‘𝑅)‘((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) / ((denom‘𝑥) · (denom‘𝑦)))))
70	30, 37	zaddcld 12847	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) ∈ ℤ)
71	36, 29	zmulcld 12849	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ)
72	64, 66, 45, 49	mulne0d 12008	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)
73	3, 14, 15	qqhvq 34730	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) ∈ ℤ ∧ ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ ∧ ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)) → ((ℚHom‘𝑅)‘((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) / ((denom‘𝑥) · (denom‘𝑦)))) = ((𝐿‘(((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
74	44, 70, 71, 72, 73	syl13anc 1400	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) / ((denom‘𝑥) · (denom‘𝑦)))) = ((𝐿‘(((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
75		rhmghm 20765	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑅) → 𝐿 ∈ (ℤ_ring GrpHom 𝑅))
76	39, 75	syl 18	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿 ∈ (ℤ_ring GrpHom 𝑅))
77		zringplusg 21814	. . . . . . 7 ⊢ + = (+_g‘ℤ_ring)
78	21, 77, 8	ghmlin 19485	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_ring GrpHom 𝑅) ∧ ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ ∧ ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ) → (𝐿‘(((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥)))) = ((𝐿‘((numer‘𝑥) · (denom‘𝑦)))(+_g‘𝑅)(𝐿‘((numer‘𝑦) · (denom‘𝑥)))))
79	78	oveq1d 7466	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_ring GrpHom 𝑅) ∧ ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ ∧ ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ) → ((𝐿‘(((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))) = (((𝐿‘((numer‘𝑥) · (denom‘𝑦)))(+_g‘𝑅)(𝐿‘((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
80	76, 30, 37, 79	syl3anc 1399	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))) = (((𝐿‘((numer‘𝑥) · (denom‘𝑦)))(+_g‘𝑅)(𝐿‘((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
81	69, 74, 80	3eqtrd 2802	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 + 𝑦)) = (((𝐿‘((numer‘𝑥) · (denom‘𝑦)))(+_g‘𝑅)(𝐿‘((numer‘𝑦) · (denom‘𝑥)))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
82	58	fveq2d 6920	. . . . . 6 ⊢ (𝑥 ∈ ℚ → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
83	82	ad2antrl 741	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
84	3, 14, 15	qqhvq 34730	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((numer‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ≠ 0)) → ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
85	44, 26, 36, 45, 84	syl13anc 1400	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
86	52, 21, 14, 40	rhmdvd 33989	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑅) ∧ ((numer‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ∈ ℤ ∧ (denom‘𝑦) ∈ ℤ) ∧ ((𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅) ∧ (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))) → ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))) = ((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
87	39, 26, 36, 29, 48, 51, 86	syl132anc 1416	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))) = ((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
88	83, 85, 87	3eqtrd 2802	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
89	60	fveq2d 6920	. . . . . 6 ⊢ (𝑦 ∈ ℚ → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
90	89	ad2antll 742	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
91	52, 21, 14, 40	rhmdvd 33989	. . . . . . 7 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑅) ∧ ((numer‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ∈ ℤ ∧ (denom‘𝑥) ∈ ℤ) ∧ ((𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅) ∧ (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))) → ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑦) · (denom‘𝑥)))))
92	39, 33, 29, 36, 51, 48, 91	syl132anc 1416	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑦) · (denom‘𝑥)))))
93	3, 14, 15	qqhvq 34730	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((numer‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ≠ 0)) → ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
94	44, 33, 29, 49, 93	syl13anc 1400	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
95	64, 66	mulcomd 11369	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) = ((denom‘𝑦) · (denom‘𝑥)))
96	95	fveq2d 6920	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = (𝐿‘((denom‘𝑦) · (denom‘𝑥))))
97	96	oveq2d 7467	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑦) · (denom‘𝑥)))))
98	92, 94, 97	3eqtr4d 2808	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
99	90, 98	eqtrd 2798	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
100	88, 99	oveq12d 7469	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((ℚHom‘𝑅)‘𝑥)(+_g‘𝑅)((ℚHom‘𝑅)‘𝑦)) = (((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦))))(+_g‘𝑅)((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦))))))
101	57, 81, 100	3eqtr4d 2808	. 2 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 + 𝑦)) = (((ℚHom‘𝑅)‘𝑥)(+_g‘𝑅)((ℚHom‘𝑅)‘𝑦)))
102	2, 3, 7, 8, 11, 13, 16, 101	isghmd 19489	1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 Vcvv 3454 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 0cc0 11239 + caddc 11242 · cmul 11244 / cdiv 12013 ℕcn 12375 ℤcz 12733 ℚcq 13115 numercnumer 16953 denomcdenom 16954 Basecbs 17431 ↾_s cress 17452 +_gcplusg 17472 ._rcmulr 17473 0_gc0g 17660 Grpcgrp 19194 GrpHom cghm 19477 Ringcrg 20510 Unitcui 20636 /_rcdvr 20681 RingHom crh 20750 DivRingcdr 21031 ℂ_fldccnfld 21729 ℤ_ringczring 21806 ℤRHomczrh 21860 chrcchr 21862 ℚHomcqqh 34713

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318 ax-mulf 11319

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-fz 13680 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-gcd 16709 df-numer 16955 df-denom 16956 df-gz 17152 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-euf 17499 df-0g 17662 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-ghm 19478 df-od 19793 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-cring 20513 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-dvr 20682 df-rhm 20753 df-subrng 20849 df-subrg 20873 df-drng 21033 df-cnfld 21730 df-zring 21807 df-zrh 21864 df-chr 21866 df-qqh 34714

This theorem is used by: qqhcn 34734 qqhucn 34735

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