qqhrhm - Mathbox for Thierry Arnoux

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Theorem qqhrhm 31839

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 26672	. 2 ⊢ ℚ = (Base‘𝑄)
3	1	qrng1 26675	. 2 ⊢ 1 = (1_r‘𝑄)
4		eqid 2738	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qex 12630	. . 3 ⊢ ℚ ∈ V
6		cnfldmul 20516	. . . 4 ⊢ · = (._r‘ℂ_fld)
7	1, 6	ressmulr 16943	. . 3 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
8	5, 7	ax-mp 5	. 2 ⊢ · = (._r‘𝑄)
9		eqid 2738	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
10	1	qdrng 26673	. . 3 ⊢ 𝑄 ∈ DivRing
11		drngring 19913	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
12	10, 11	mp1i 13	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → 𝑄 ∈ Ring)
13		isfld 19915	. . . . 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 19913	. . 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 31835	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
22	14, 21	sylan 579	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1_r‘𝑅))
23		eqid 2738	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
24		eqid 2738	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
25	13	simprbi 496	. . . . 5 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
26	25	ad2antrr 722	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ CRing)
27	20	zrhrhm 20625	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑅))
28		zringbas 20588	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
29	28, 18	rhmf 19885	. . . . . . 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 16372	. . . . . 6 ⊢ (𝑥 ∈ ℚ → (numer‘𝑥) ∈ ℤ)
33	32	ad2antrl 724	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℤ)
34	31, 33	ffvelrnd 6944	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑥)) ∈ 𝐵)
35	14	ad2antrr 722	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑅 ∈ DivRing)
36		simplr 765	. . . . . 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 16373	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → (denom‘𝑥) ∈ ℕ)
39	38	ad2antrl 724	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℕ)
40	39	nnzd 12354	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℤ)
41	39	nnne0d 11953	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ≠ 0)
42		eqid 2738	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
43	18, 20, 42	elzrhunit 31829	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑥) ∈ ℤ ∧ (denom‘𝑥) ≠ 0)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
44	37, 40, 41, 43	syl12anc 833	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(denom‘𝑥)) ∈ (Unit‘𝑅))
45		qnumcl 16372	. . . . . 6 ⊢ (𝑦 ∈ ℚ → (numer‘𝑦) ∈ ℤ)
46	45	ad2antll 725	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℤ)
47	31, 46	ffvelrnd 6944	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘(numer‘𝑦)) ∈ 𝐵)
48		qdencl 16373	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → (denom‘𝑦) ∈ ℕ)
49	48	ad2antll 725	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℕ)
50	49	nnzd 12354	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℤ)
51	49	nnne0d 11953	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ≠ 0)
52	18, 20, 42	elzrhunit 31829	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ((denom‘𝑦) ∈ ℤ ∧ (denom‘𝑦) ≠ 0)) → (𝐿‘(denom‘𝑦)) ∈ (Unit‘𝑅))
53	37, 50, 51, 52	syl12anc 833	. . . 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 31390	. . 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 16378	. . . . . . 7 ⊢ (𝑥 ∈ ℚ → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
56	55	fveq2d 6760	. . . . . 6 ⊢ (𝑥 ∈ ℚ → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
57	56	ad2antrl 724	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((ℚHom‘𝑅)‘((numer‘𝑥) / (denom‘𝑥))))
58	18, 19, 20	qqhvq 31837	. . . . . 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 1370	. . . . 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 2778	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑥) = ((𝐿‘(numer‘𝑥)) / (𝐿‘(denom‘𝑥))))
61		qeqnumdivden 16378	. . . . . . 7 ⊢ (𝑦 ∈ ℚ → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
62	61	fveq2d 6760	. . . . . 6 ⊢ (𝑦 ∈ ℚ → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
63	62	ad2antll 725	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((ℚHom‘𝑅)‘((numer‘𝑦) / (denom‘𝑦))))
64	18, 19, 20	qqhvq 31837	. . . . . 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 1370	. . . . 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 2778	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘𝑦) = ((𝐿‘(numer‘𝑦)) / (𝐿‘(denom‘𝑦))))
67	60, 66	oveq12d 7273	. . 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 724	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑥 = ((numer‘𝑥) / (denom‘𝑥)))
69	61	ad2antll 725	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → 𝑦 = ((numer‘𝑦) / (denom‘𝑦)))
70	68, 69	oveq12d 7273	. . . . . 6 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) / (denom‘𝑥)) · ((numer‘𝑦) / (denom‘𝑦))))
71	33	zcnd 12356	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑥) ∈ ℂ)
72	40	zcnd 12356	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑥) ∈ ℂ)
73	46	zcnd 12356	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (numer‘𝑦) ∈ ℂ)
74	50	zcnd 12356	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (denom‘𝑦) ∈ ℂ)
75	71, 72, 73, 74, 41, 51	divmuldivd 11722	. . . . . 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 2778	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 · 𝑦) = (((numer‘𝑥) · (numer‘𝑦)) / ((denom‘𝑥) · (denom‘𝑦))))
77	76	fveq2d 6760	. . . 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 12361	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (numer‘𝑦)) ∈ ℤ)
79	40, 50	zmulcld 12361	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ∈ ℤ)
80	72, 74, 41, 51	mulne0d 11557	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) ≠ 0)
81	18, 19, 20	qqhvq 31837	. . . . 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 1370	. . . 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 20591	. . . . . . 7 ⊢ · = (._r‘ℤ_ring)
86	28, 85, 9	rhmmul 19886	. . . . . 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 1369	. . . . 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 19886	. . . . . 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 1369	. . . . 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 7273	. . . 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 2782	. . 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 2789	. 2 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((ℚHom‘𝑅)‘(𝑥 · 𝑦)) = (((ℚHom‘𝑅)‘𝑥)(._r‘𝑅)((ℚHom‘𝑅)‘𝑦)))
93		cnfldadd 20515	. . . 4 ⊢ + = (+_g‘ℂ_fld)
94	1, 93	ressplusg 16926	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
95	5, 94	ax-mp 5	. 2 ⊢ + = (+_g‘𝑄)
96	18, 19, 20	qqhf 31836	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
97	14, 96	sylan 579	. 2 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
98	33, 50	zmulcld 12361	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑥) · (denom‘𝑦)) ∈ ℤ)
99	31, 98	ffvelrnd 6944	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑥) · (denom‘𝑦))) ∈ 𝐵)
100	46, 40	zmulcld 12361	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((numer‘𝑦) · (denom‘𝑥)) ∈ ℤ)
101	31, 100	ffvelrnd 6944	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((numer‘𝑦) · (denom‘𝑥))) ∈ 𝐵)
102	23, 9	unitmulcl 19821	. . . . . 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 1369	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((𝐿‘(denom‘𝑥))(._r‘𝑅)(𝐿‘(denom‘𝑦))) ∈ (Unit‘𝑅))
104	89, 103	eqeltrd 2839	. . . 4 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) ∈ (Unit‘𝑅))
105	18, 23, 24, 19	dvrdir 31389	. . . 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 1370	. . 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 7273	. . . . . 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 11725	. . . . . 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 2778	. . . . 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 6760	. . . 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 12359	. . . . 5 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (((numer‘𝑥) · (denom‘𝑦)) + ((numer‘𝑦) · (denom‘𝑥))) ∈ ℤ)
112	18, 19, 20	qqhvq 31837	. . . . 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 1370	. . . 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 19884	. . . . . 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 20589	. . . . . . 7 ⊢ + = (+_g‘ℤ_ring)
117	28, 116, 24	ghmlin 18754	. . . . . 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 7270	. . . . 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 1369	. . . 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 2782	. . 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 31422	. . . . . 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 1386	. . . . 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 2782	. . . 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 31422	. . . . . . 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 1386	. . . . . 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 10927	. . . . . . . 8 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → ((denom‘𝑥) · (denom‘𝑦)) = ((denom‘𝑦) · (denom‘𝑥)))
127	126	fveq2d 6760	. . . . . . 7 ⊢ (((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝐿‘((denom‘𝑥) · (denom‘𝑦))) = (𝐿‘((denom‘𝑦) · (denom‘𝑥))))
128	127	oveq2d 7271	. . . . . 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 2788	. . . . 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 2778	. . . 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 7273	. . 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 2788	. 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 19888	1 ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 / cdiv 11562 ℕcn 11903 ℤcz 12249 ℚcq 12617 numercnumer 16365 denomcdenom 16366 Basecbs 16840 ↾_s cress 16867 +_gcplusg 16888 ._rcmulr 16889 0_gc0g 17067 GrpHom cghm 18746 1_rcur 19652 Ringcrg 19698 CRingccrg 19699 Unitcui 19796 /_rcdvr 19839 RingHom crh 19871 DivRingcdr 19906 Fieldcfield 19907 ℂ_fldccnfld 20510 ℤ_ringzring 20582 ℤRHomczrh 20613 chrcchr 20615 ℚHomcqqh 31822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-fz 13169 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-numer 16367 df-denom 16368 df-gz 16559 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-od 19051 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-field 19909 df-subrg 19937 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-chr 20619 df-qqh 31823

This theorem is referenced by: (None)

W3C validator