qqhrhm - Mathbox for Thierry Arnoux

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Theorem qqhrhm 34379

Description: The ℚHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-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
qqhrhm	⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))

Proof of Theorem qqhrhm 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 27783	. 2 ⊢ ℚ = (Base‘𝑄)
3	1	qrng1 27786	. 2 ⊢ 1 = (1_r‘𝑄)
4		eqid 2763	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qex 12980	. . 3 ⊢ ℚ ∈ V
6		cnfldmul 21530	. . . 4 ⊢ · = (._r‘ℂ_fld)
7	1, 6	ressmulr 17355	. . 3 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
8	5, 7	ax-mp 5	. 2 ⊢ · = (._r‘𝑄)
9		eqid 2763	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
10	1	qdrng 27784	. . 3 ⊢ 𝑄 ∈ DivRing
11		drngring 20834	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
12	10, 11	mp1i 14	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑄 ∈ Ring)
13		isfld 20840	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
14	13	simplbi 501	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
15	14	adantr 485	. . 3 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
16		drngring 20834	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
17	15, 16	syl 18	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑅 ∈ Ring)
18		qqhval2.0	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
19		qqhval2.1	. . . 4 ⊢ / = (/_r‘𝑅)
20		qqhval2.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅)
21	18, 19, 20	qqh1 34375	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
22	14, 21	sylan 591	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
23		eqid 2763	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
24		eqid 2763	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
25	13	simprbi 502	. . . . 5 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
26	25	ad2antrr 738	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ CRing)
27	20	zrhrhm 21661	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
28		zringbas 21603	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
29	28, 18	rhmf 20563	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑅) → 𝐿:ℤ⟶𝐵)
30	17, 27, 29	3syl 19	. . . . . 6 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝐿:ℤ⟶𝐵)
31	30	adantr 485	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿:ℤ⟶𝐵)
32		qnumcl 16794	. . . . . 6 ⊢ (𝑥 ∈ ℚ → (numer‘𝑥) ∈ ℤ)
33	32	ad2antrl 740	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℤ)
34	31, 33	ffvelcdmd 7080	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑥)) ∈ 𝐵)
35	14	ad2antrr 738	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ DivRing)
36		simplr 780	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (chr‘𝑅) = 0)
37	35, 36	jca 520	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0))
38		qdencl 16795	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → (denom‘𝑥) ∈ ℕ)
39	38	ad2antrl 740	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℕ)
40	39	nnzd 12612	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℤ)
41	39	nnne0d 12281	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ≠ 0)
42		eqid 2763	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
43	18, 20, 42	elzrhunit 34367	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ≠ 0)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
44	37, 40, 41, 43	syl12anc 849	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
45		qnumcl 16794	. . . . . 6 ⊢ (𝑦 ∈ ℚ → (numer‘𝑦) ∈ ℤ)
46	45	ad2antll 741	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℤ)
47	31, 46	ffvelcdmd 7080	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑦)) ∈ 𝐵)
48		qdencl 16795	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (denom‘𝑦) ∈ ℕ)
49	48	ad2antll 741	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℕ)
50	49	nnzd 12612	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℤ)
51	49	nnne0d 12281	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ≠ 0)
52	18, 20, 42	elzrhunit 34367	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ≠ 0)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
53	37, 50, 51, 52	syl12anc 849	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
54	18, 23, 24, 19, 9, 26, 34, 44, 47, 53	rdivmuldivd 20491	. . 3 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥)))(._r‘𝑅)((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦)))) = (((𝐿‘(numer‘𝑥))(._r‘𝑅)(𝐿‘(numer‘𝑦))) / ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦)))))
55		qeqnumdivden 16800	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
56	55	fveq2d 6885	. . . . . 6 ⊢ (𝑥 ∈ ℚ → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
57	56	ad2antrl 740	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
58	18, 19, 20	qqhvq 34377	. . . . . 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))))
59	37, 33, 40, 41, 58	syl13anc 1399	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
60	57, 59	eqtrd 2798	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
61		qeqnumdivden 16800	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
62	61	fveq2d 6885	. . . . . 6 ⊢ (𝑦 ∈ ℚ → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
63	62	ad2antll 741	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
64	18, 19, 20	qqhvq 34377	. . . . . 6 ⊢ (((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))))
65	37, 46, 50, 51, 64	syl13anc 1399	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
66	63, 65	eqtrd 2798	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
67	60, 66	oveq12d 7428	. . 3 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((ℚHom‘𝑅)‘𝑥)(._r‘𝑅)((ℚHom‘𝑅)‘𝑦)) = (((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥)))(._r‘𝑅)((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦)))))
68	55	ad2antrl 740	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
69	61	ad2antll 741	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
70	68, 69	oveq12d 7428	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) / (denom‘𝑥)) · ((numer‘𝑦) / (denom‘𝑦))))
71	33	zcnd 12696	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℂ)
72	40	zcnd 12696	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℂ)
73	46	zcnd 12696	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℂ)
74	50	zcnd 12696	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℂ)
75	71, 72, 73, 74, 41, 51	divmuldivd 12027	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) / (denom‘𝑥)) · ((numer‘𝑦) / (denom‘𝑦))) = (((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦))))
76	70, 75	eqtrd 2798	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦))))
77	76	fveq2d 6885	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 · 𝑦)) = ((ℚHom‘𝑅)‘(((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦)))))
78	33, 46	zmulcld 12701	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (numer‘𝑦)) ∈ ℤ)
79	40, 50	zmulcld 12701	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ)
80	72, 74, 41, 51	mulne0d 11861	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)
81	18, 19, 20	qqhvq 34377	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (((numer‘𝑥) · (numer‘𝑦)) ∈ ℤ ∧ ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ ∧ ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)) → ((ℚHom‘𝑅)‘(((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦)))) = ((𝐿‘((numer‘𝑥) · (numer‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
82	37, 78, 79, 80, 81	syl13anc 1399	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦)))) = ((𝐿‘((numer‘𝑥) · (numer‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
83	35, 16	syl 18	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ Ring)
84	83, 27	syl 18	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
85		zringmulr 21607	. . . . . . 7 ⊢ · = (._r‘ℤ_ring)
86	28, 85, 9	rhmmul 20568	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑅) ∧ (numer‘𝑥) ∈ ℤ ∧ (numer‘𝑦) ∈ ℤ) → (𝐿‘((numer‘𝑥) · (numer‘𝑦))) = ((𝐿‘(numer‘𝑥))(._r‘𝑅)(𝐿‘(numer‘𝑦))))
87	84, 33, 46, 86	syl3anc 1398	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑥) · (numer‘𝑦))) = ((𝐿‘(numer‘𝑥))(._r‘𝑅)(𝐿‘(numer‘𝑦))))
88	28, 85, 9	rhmmul 20568	. . . . . 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))))
89	84, 40, 50, 88	syl3anc 1398	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))))
90	87, 89	oveq12d 7428	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘((numer‘𝑥) · (numer‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))) = (((𝐿‘(numer‘𝑥))(._r‘𝑅)(𝐿‘(numer‘𝑦))) / ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦)))))
91	77, 82, 90	3eqtrd 2802	. . 3 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 · 𝑦)) = (((𝐿‘(numer‘𝑥))(._r‘𝑅)(𝐿‘(numer‘𝑦))) / ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦)))))
92	54, 67, 91	3eqtr4rd 2809	. 2 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 · 𝑦)) = (((ℚHom‘𝑅)‘𝑥)(._r‘𝑅)((ℚHom‘𝑅)‘𝑦)))
93		cnfldadd 21528	. . . 4 ⊢ + = (+_g‘ℂ_fld)
94	1, 93	ressplusg 17339	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
95	5, 94	ax-mp 5	. 2 ⊢ + = (+_g‘𝑄)
96	18, 19, 20	qqhf 34376	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
97	14, 96	sylan 591	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
98	33, 50	zmulcld 12701	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ)
99	31, 98	ffvelcdmd 7080	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑥) · (denom‘𝑦))) ∈ 𝐵)
100	46, 40	zmulcld 12701	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ)
101	31, 100	ffvelcdmd 7080	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑦) · (denom‘𝑥))) ∈ 𝐵)
102	23, 9	unitmulcl 20458	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅) ∧ (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
103	83, 44, 53, 102	syl3anc 1398	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
104	89, 103	eqeltrd 2863	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) ∈ (Unit‘𝑅))
105	18, 23, 24, 19	dvrdir 20490	. . . 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))))))
106	83, 99, 101, 104, 105	syl13anc 1399	. . 3 ⊢ (((𝑅 ∈ Field ∧ (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))))))
107	68, 69	oveq12d 7428	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) = (((numer‘𝑥) / (denom‘𝑥)) + ((numer‘𝑦) / (denom‘𝑦))))
108	71, 72, 73, 74, 41, 51	divadddivd 12030	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (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))))
109	107, 108	eqtrd 2798	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) = ((((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) / ((denom‘𝑥) · (denom‘𝑦))))
110	109	fveq2d 6885	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (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)))))
111	98, 100	zaddcld 12699	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) ∈ ℤ)
112	18, 19, 20	qqhvq 34377	. . . . 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)))))
113	37, 111, 79, 80, 112	syl13anc 1399	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (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)))))
114		rhmghm 20562	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑅) → 𝐿 ∈ (ℤ_ring GrpHom 𝑅))
115	84, 114	syl 18	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿 ∈ (ℤ_ring GrpHom 𝑅))
116		zringplusg 21604	. . . . . . 7 ⊢ + = (+_g‘ℤ_ring)
117	28, 116, 24	ghmlin 19286	. . . . . 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)))))
118	117	oveq1d 7425	. . . . 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)))))
119	115, 98, 100, 118	syl3anc 1398	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (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)))))
120	110, 113, 119	3eqtrd 2802	. . 3 ⊢ (((𝑅 ∈ Field ∧ (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)))))
121	23, 28, 19, 85	rhmdvd 33644	. . . . . 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)))))
122	84, 33, 40, 50, 44, 53, 121	syl132anc 1415	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))) = ((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
123	57, 59, 122	3eqtrd 2802	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((𝐿‘((numer‘𝑥) · (denom‘𝑦))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
124	23, 28, 19, 85	rhmdvd 33644	. . . . . . 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)))))
125	84, 46, 50, 40, 53, 44, 124	syl132anc 1415	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑦) · (denom‘𝑥)))))
126	72, 74	mulcomd 11225	. . . . . . . 8 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) = ((denom‘𝑦) · (denom‘𝑥)))
127	126	fveq2d 6885	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = (𝐿‘((denom‘𝑦) · (denom‘𝑥))))
128	127	oveq2d 7426	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (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)))))
129	125, 65, 128	3eqtr4d 2808	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
130	63, 129	eqtrd 2798	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((𝐿‘((numer‘𝑦) · (denom‘𝑥))) / (𝐿‘((denom‘𝑥) · (denom‘𝑦)))))
131	123, 130	oveq12d 7428	. . 3 ⊢ (((𝑅 ∈ Field ∧ (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))))))
132	106, 120, 131	3eqtr4d 2808	. 2 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 + 𝑦)) = (((ℚHom‘𝑅)‘𝑥)(+_g‘𝑅)((ℚHom‘𝑅)‘𝑦)))
133	2, 3, 4, 8, 9, 12, 17, 22, 92, 18, 95, 24, 97, 132	isrhmd 20570	1 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 / cdiv 11866 ℕcn 12228 ℤcz 12586 ℚcq 12967 numercnumer 16787 denomcdenom 16788 Basecbs 17264 ↾_s cress 17285 +_gcplusg 17305 ._rcmulr 17306 0_gc0g 17487 GrpHom cghm 19278 1_rcur 20258 Ringcrg 20310 CRingccrg 20311 Unitcui 20433 /_rcdvr 20478 RingHom crh 20547 DivRingcdr 20827 Fieldcfield 20828 ℂ_fldccnfld 21522 ℤ_ringczring 21596 ℤRHomczrh 21649 chrcchr 21651 ℚHomcqqh 34360

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-fz 13531 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-gcd 16548 df-numer 16789 df-denom 16790 df-gz 16985 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-0g 17489 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-grp 18998 df-minusg 18999 df-sbg 19000 df-mulg 19129 df-subg 19184 df-ghm 19279 df-od 19593 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-rhm 20550 df-subrng 20645 df-subrg 20669 df-drng 20829 df-field 20830 df-cnfld 21523 df-zring 21597 df-zrh 21653 df-chr 21655 df-qqh 34361

This theorem is referenced by: (None)

W3C validator