qqhrhm - Mathbox for Thierry Arnoux

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Theorem qqhrhm 33986

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 27537	. 2 ⊢ ℚ = (Base‘𝑄)
3	1	qrng1 27540	. 2 ⊢ 1 = (1_r‘𝑄)
4		eqid 2730	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qex 12927	. . 3 ⊢ ℚ ∈ V
6		cnfldmul 21279	. . . 4 ⊢ · = (._r‘ℂ_fld)
7	1, 6	ressmulr 17277	. . 3 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
8	5, 7	ax-mp 5	. 2 ⊢ · = (._r‘𝑄)
9		eqid 2730	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
10	1	qdrng 27538	. . 3 ⊢ 𝑄 ∈ DivRing
11		drngring 20652	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
12	10, 11	mp1i 13	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑄 ∈ Ring)
13		isfld 20656	. . . . 5 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
14	13	simplbi 497	. . . 4 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing)
15	14	adantr 480	. . 3 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
16		drngring 20652	. . 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 33982	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
22	14, 21	sylan 580	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
23		eqid 2730	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
24		eqid 2730	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
25	13	simprbi 496	. . . . 5 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
26	25	ad2antrr 726	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ CRing)
27	20	zrhrhm 21428	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
28		zringbas 21370	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
29	28, 18	rhmf 20401	. . . . . . 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 16717	. . . . . 6 ⊢ (𝑥 ∈ ℚ → (numer‘𝑥) ∈ ℤ)
33	32	ad2antrl 728	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℤ)
34	31, 33	ffvelcdmd 7060	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑥)) ∈ 𝐵)
35	14	ad2antrr 726	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ DivRing)
36		simplr 768	. . . . . 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 16718	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → (denom‘𝑥) ∈ ℕ)
39	38	ad2antrl 728	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℕ)
40	39	nnzd 12563	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℤ)
41	39	nnne0d 12243	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ≠ 0)
42		eqid 2730	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
43	18, 20, 42	elzrhunit 33974	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ≠ 0)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
44	37, 40, 41, 43	syl12anc 836	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
45		qnumcl 16717	. . . . . 6 ⊢ (𝑦 ∈ ℚ → (numer‘𝑦) ∈ ℤ)
46	45	ad2antll 729	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℤ)
47	31, 46	ffvelcdmd 7060	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑦)) ∈ 𝐵)
48		qdencl 16718	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (denom‘𝑦) ∈ ℕ)
49	48	ad2antll 729	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℕ)
50	49	nnzd 12563	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℤ)
51	49	nnne0d 12243	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ≠ 0)
52	18, 20, 42	elzrhunit 33974	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ≠ 0)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
53	37, 50, 51, 52	syl12anc 836	. . . 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 20329	. . 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 16723	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
56	55	fveq2d 6865	. . . . . 6 ⊢ (𝑥 ∈ ℚ → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
57	56	ad2antrl 728	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
58	18, 19, 20	qqhvq 33984	. . . . . 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 1374	. . . . 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 2765	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
61		qeqnumdivden 16723	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
62	61	fveq2d 6865	. . . . . 6 ⊢ (𝑦 ∈ ℚ → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
63	62	ad2antll 729	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
64	18, 19, 20	qqhvq 33984	. . . . . 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 1374	. . . . 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 2765	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
67	60, 66	oveq12d 7408	. . 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 728	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
69	61	ad2antll 729	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
70	68, 69	oveq12d 7408	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) / (denom‘𝑥)) · ((numer‘𝑦) / (denom‘𝑦))))
71	33	zcnd 12646	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℂ)
72	40	zcnd 12646	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℂ)
73	46	zcnd 12646	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℂ)
74	50	zcnd 12646	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℂ)
75	71, 72, 73, 74, 41, 51	divmuldivd 12006	. . . . . 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 2765	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦))))
77	76	fveq2d 6865	. . . 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 12651	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (numer‘𝑦)) ∈ ℤ)
79	40, 50	zmulcld 12651	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ)
80	72, 74, 41, 51	mulne0d 11837	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)
81	18, 19, 20	qqhvq 33984	. . . . 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 1374	. . . 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 21374	. . . . . . 7 ⊢ · = (._r‘ℤ_ring)
86	28, 85, 9	rhmmul 20402	. . . . . 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 1373	. . . . 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 20402	. . . . . 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 1373	. . . . 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 7408	. . . 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 2769	. . 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 2776	. 2 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 · 𝑦)) = (((ℚHom‘𝑅)‘𝑥)(._r‘𝑅)((ℚHom‘𝑅)‘𝑦)))
93		cnfldadd 21277	. . . 4 ⊢ + = (+_g‘ℂ_fld)
94	1, 93	ressplusg 17261	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
95	5, 94	ax-mp 5	. 2 ⊢ + = (+_g‘𝑄)
96	18, 19, 20	qqhf 33983	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
97	14, 96	sylan 580	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
98	33, 50	zmulcld 12651	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ)
99	31, 98	ffvelcdmd 7060	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑥) · (denom‘𝑦))) ∈ 𝐵)
100	46, 40	zmulcld 12651	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ)
101	31, 100	ffvelcdmd 7060	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑦) · (denom‘𝑥))) ∈ 𝐵)
102	23, 9	unitmulcl 20296	. . . . . 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 1373	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
104	89, 103	eqeltrd 2829	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) ∈ (Unit‘𝑅))
105	18, 23, 24, 19	dvrdir 20328	. . . 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 1374	. . 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 7408	. . . . . 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 12009	. . . . . 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 2765	. . . . 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 6865	. . . 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 12649	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) ∈ ℤ)
112	18, 19, 20	qqhvq 33984	. . . . 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 1374	. . . 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 20400	. . . . . 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 21371	. . . . . . 7 ⊢ + = (+_g‘ℤ_ring)
117	28, 116, 24	ghmlin 19160	. . . . . 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 7405	. . . . 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 1373	. . . 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 2769	. . 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 33303	. . . . . 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 1390	. . . . 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 2769	. . . 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 33303	. . . . . . 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 1390	. . . . . 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 11202	. . . . . . . 8 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) = ((denom‘𝑦) · (denom‘𝑥)))
127	126	fveq2d 6865	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = (𝐿‘((denom‘𝑦) · (denom‘𝑥))))
128	127	oveq2d 7406	. . . . . 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 2775	. . . . 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 2765	. . . 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 7408	. . 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 2775	. 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 20404	1 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 Vcvv 3450 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 / cdiv 11842 ℕcn 12193 ℤcz 12536 ℚcq 12914 numercnumer 16710 denomcdenom 16711 Basecbs 17186 ↾_s cress 17207 +_gcplusg 17227 ._rcmulr 17228 0_gc0g 17409 GrpHom cghm 19151 1_rcur 20097 Ringcrg 20149 CRingccrg 20150 Unitcui 20271 /_rcdvr 20316 RingHom crh 20385 DivRingcdr 20645 Fieldcfield 20646 ℂ_fldccnfld 21271 ℤ_ringczring 21363 ℤRHomczrh 21416 chrcchr 21418 ℚHomcqqh 33967

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-fz 13476 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472 df-numer 16712 df-denom 16713 df-gz 16908 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-ghm 19152 df-od 19465 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20462 df-subrg 20486 df-drng 20647 df-field 20648 df-cnfld 21272 df-zring 21364 df-zrh 21420 df-chr 21422 df-qqh 33968

This theorem is referenced by: (None)

W3C validator