qqhrhm - Mathbox for Thierry Arnoux

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Theorem qqhrhm 34171

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 27601	. 2 ⊢ ℚ = (Base‘𝑄)
3	1	qrng1 27604	. 2 ⊢ 1 = (1_r‘𝑄)
4		eqid 2737	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qex 12886	. . 3 ⊢ ℚ ∈ V
6		cnfldmul 21332	. . . 4 ⊢ · = (._r‘ℂ_fld)
7	1, 6	ressmulr 17239	. . 3 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
8	5, 7	ax-mp 5	. 2 ⊢ · = (._r‘𝑄)
9		eqid 2737	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
10	1	qdrng 27602	. . 3 ⊢ 𝑄 ∈ DivRing
11		drngring 20684	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
12	10, 11	mp1i 13	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑄 ∈ Ring)
13		isfld 20688	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
14	13	simplbi 496	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
15	14	adantr 480	. . 3 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
16		drngring 20684	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
17	15, 16	syl 17	. 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 34167	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
22	14, 21	sylan 581	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
23		eqid 2737	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
24		eqid 2737	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
25	13	simprbi 497	. . . . 5 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
26	25	ad2antrr 727	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ CRing)
27	20	zrhrhm 21481	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
28		zringbas 21423	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
29	28, 18	rhmf 20435	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑅) → 𝐿:ℤ⟶𝐵)
30	17, 27, 29	3syl 18	. . . . . 6 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝐿:ℤ⟶𝐵)
31	30	adantr 480	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿:ℤ⟶𝐵)
32		qnumcl 16679	. . . . . 6 ⊢ (𝑥 ∈ ℚ → (numer‘𝑥) ∈ ℤ)
33	32	ad2antrl 729	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℤ)
34	31, 33	ffvelcdmd 7039	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑥)) ∈ 𝐵)
35	14	ad2antrr 727	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ DivRing)
36		simplr 769	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (chr‘𝑅) = 0)
37	35, 36	jca 511	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0))
38		qdencl 16680	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → (denom‘𝑥) ∈ ℕ)
39	38	ad2antrl 729	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℕ)
40	39	nnzd 12526	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℤ)
41	39	nnne0d 12207	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ≠ 0)
42		eqid 2737	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
43	18, 20, 42	elzrhunit 34159	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ≠ 0)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
44	37, 40, 41, 43	syl12anc 837	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
45		qnumcl 16679	. . . . . 6 ⊢ (𝑦 ∈ ℚ → (numer‘𝑦) ∈ ℤ)
46	45	ad2antll 730	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℤ)
47	31, 46	ffvelcdmd 7039	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑦)) ∈ 𝐵)
48		qdencl 16680	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (denom‘𝑦) ∈ ℕ)
49	48	ad2antll 730	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℕ)
50	49	nnzd 12526	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℤ)
51	49	nnne0d 12207	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ≠ 0)
52	18, 20, 42	elzrhunit 34159	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ≠ 0)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
53	37, 50, 51, 52	syl12anc 837	. . . 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 20364	. . 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 16685	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
56	55	fveq2d 6846	. . . . . 6 ⊢ (𝑥 ∈ ℚ → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
57	56	ad2antrl 729	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
58	18, 19, 20	qqhvq 34169	. . . . . 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 1375	. . . . 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 2772	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
61		qeqnumdivden 16685	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
62	61	fveq2d 6846	. . . . . 6 ⊢ (𝑦 ∈ ℚ → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
63	62	ad2antll 730	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
64	18, 19, 20	qqhvq 34169	. . . . . 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 1375	. . . . 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 2772	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
67	60, 66	oveq12d 7386	. . 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 729	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
69	61	ad2antll 730	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
70	68, 69	oveq12d 7386	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) / (denom‘𝑥)) · ((numer‘𝑦) / (denom‘𝑦))))
71	33	zcnd 12609	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℂ)
72	40	zcnd 12609	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℂ)
73	46	zcnd 12609	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℂ)
74	50	zcnd 12609	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℂ)
75	71, 72, 73, 74, 41, 51	divmuldivd 11970	. . . . . 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 2772	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦))))
77	76	fveq2d 6846	. . . 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 12614	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (numer‘𝑦)) ∈ ℤ)
79	40, 50	zmulcld 12614	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ)
80	72, 74, 41, 51	mulne0d 11801	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)
81	18, 19, 20	qqhvq 34169	. . . . 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 1375	. . . 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 17	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ Ring)
84	83, 27	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
85		zringmulr 21427	. . . . . . 7 ⊢ · = (._r‘ℤ_ring)
86	28, 85, 9	rhmmul 20436	. . . . . 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 1374	. . . . 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 20436	. . . . . 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 1374	. . . . 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 7386	. . . 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 2776	. . 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 2783	. 2 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 · 𝑦)) = (((ℚHom‘𝑅)‘𝑥)(._r‘𝑅)((ℚHom‘𝑅)‘𝑦)))
93		cnfldadd 21330	. . . 4 ⊢ + = (+_g‘ℂ_fld)
94	1, 93	ressplusg 17223	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
95	5, 94	ax-mp 5	. 2 ⊢ + = (+_g‘𝑄)
96	18, 19, 20	qqhf 34168	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
97	14, 96	sylan 581	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
98	33, 50	zmulcld 12614	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ)
99	31, 98	ffvelcdmd 7039	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑥) · (denom‘𝑦))) ∈ 𝐵)
100	46, 40	zmulcld 12614	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ)
101	31, 100	ffvelcdmd 7039	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑦) · (denom‘𝑥))) ∈ 𝐵)
102	23, 9	unitmulcl 20331	. . . . . 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 1374	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
104	89, 103	eqeltrd 2837	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) ∈ (Unit‘𝑅))
105	18, 23, 24, 19	dvrdir 20363	. . . 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 1375	. . 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 7386	. . . . . 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 11973	. . . . . 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 2772	. . . . 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 6846	. . . 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 12612	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) ∈ ℤ)
112	18, 19, 20	qqhvq 34169	. . . . 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 1375	. . . 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 20434	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑅) → 𝐿 ∈ (ℤ_ring GrpHom 𝑅))
115	84, 114	syl 17	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝐿 ∈ (ℤ_ring GrpHom 𝑅))
116		zringplusg 21424	. . . . . . 7 ⊢ + = (+_g‘ℤ_ring)
117	28, 116, 24	ghmlin 19165	. . . . . 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 7383	. . . . 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 1374	. . . 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 2776	. . 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 33421	. . . . . 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 1391	. . . . 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 2776	. . . 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 33421	. . . . . . 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 1391	. . . . . 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 11165	. . . . . . . 8 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) = ((denom‘𝑦) · (denom‘𝑥)))
127	126	fveq2d 6846	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = (𝐿‘((denom‘𝑦) · (denom‘𝑥))))
128	127	oveq2d 7384	. . . . . 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 2782	. . . . 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 2772	. . . 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 7386	. . 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 2782	. 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 20438	1 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 / cdiv 11806 ℕcn 12157 ℤcz 12500 ℚcq 12873 numercnumer 16672 denomcdenom 16673 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 ._rcmulr 17190 0_gc0g 17371 GrpHom cghm 19156 1_rcur 20131 Ringcrg 20183 CRingccrg 20184 Unitcui 20306 /_rcdvr 20351 RingHom crh 20420 DivRingcdr 20677 Fieldcfield 20678 ℂ_fldccnfld 21324 ℤ_ringczring 21416 ℤRHomczrh 21469 chrcchr 21471 ℚHomcqqh 34152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-fz 13436 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-dvds 16192 df-gcd 16434 df-numer 16674 df-denom 16675 df-gz 16870 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-0g 17373 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-grp 18881 df-minusg 18882 df-sbg 18883 df-mulg 19013 df-subg 19068 df-ghm 19157 df-od 19472 df-cmn 19726 df-abl 19727 df-mgp 20091 df-rng 20103 df-ur 20132 df-ring 20185 df-cring 20186 df-oppr 20288 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrng 20494 df-subrg 20518 df-drng 20679 df-field 20680 df-cnfld 21325 df-zring 21417 df-zrh 21473 df-chr 21475 df-qqh 34153

This theorem is referenced by: (None)

W3C validator