Theorem qqhval 32849

Description: Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Hypotheses

Ref	Expression
qqhval.1	⊢ / = (/r‘𝑅)
qqhval.2	⊢ 1 = (1r‘𝑅)
qqhval.3	⊢ 𝐿 = (ℤRHom‘𝑅)

Assertion

Ref	Expression
qqhval	⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))

Distinct variable groups:   𝑥,𝑦,𝑅   𝑦,𝐿

Allowed substitution hints:   / (𝑥,𝑦)   1 (𝑥,𝑦)   𝐿(𝑥)

Proof of Theorem qqhval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2733	. . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ)
2		fveq2 6879	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3		qqhval.3	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅)
4	2, 3	eqtr4di 2790	. . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
5	4	cnveqd 5868	. . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿)
6		fveq2 6879	. . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
7	5, 6	imaeq12d 6051	. . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅)))
8		fveq2 6879	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅))
9		qqhval.1	. . . . . . 7 ⊢ / = (/r‘𝑅)
10	8, 9	eqtr4di 2790	. . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / )
11	4	fveq1d 6881	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥))
12	4	fveq1d 6881	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦))
13	10, 11, 12	oveq123d 7415	. . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦)))
14	13	opeq2d 4874	. . . 4 ⊢ (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
15	1, 7, 14	mpoeq123dv 7469	. . 3 ⊢ (𝑓 = 𝑅 → (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
16	15	rneqd 5930	. 2 ⊢ (𝑓 = 𝑅 → ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
17		df-qqh 32848	. 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18		zex 12551	. . . 4 ⊢ ℤ ∈ V
19	3	fvexi 6893	. . . . . 6 ⊢ 𝐿 ∈ V
20	19	cnvex 7900	. . . . 5 ⊢ ◡𝐿 ∈ V
21		imaexg 7890	. . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V)
22	20, 21	ax-mp 5	. . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V
23	18, 22	mpoex 8050	. . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
24	23	rnex 7887	. 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
25	16, 17, 24	fvmpt 6985	1 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⟨cop 4629  ^◡ccnv 5669  ran crn 5671   “ cima 5673  ‘cfv 6533  (class class class)co 7394   ∈ cmpo 7396   / cdiv 11855  ℤcz 12542  1_rcur 19965  Unitcui 20123  /_rcdvr 20166  ℤRHomczrh 20984  ℚHomcqqh 32847

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-neg 11431  df-z 12543  df-qqh 32848

This theorem is referenced by:  qqhval2  32857