Theorem qqhval 33969

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 2731	. . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ)
2		fveq2 6861	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3		qqhval.3	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅)
4	2, 3	eqtr4di 2783	. . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
5	4	cnveqd 5842	. . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿)
6		fveq2 6861	. . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
7	5, 6	imaeq12d 6035	. . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅)))
8		fveq2 6861	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅))
9		qqhval.1	. . . . . . 7 ⊢ / = (/r‘𝑅)
10	8, 9	eqtr4di 2783	. . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / )
11	4	fveq1d 6863	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥))
12	4	fveq1d 6863	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦))
13	10, 11, 12	oveq123d 7411	. . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦)))
14	13	opeq2d 4847	. . . 4 ⊢ (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
15	1, 7, 14	mpoeq123dv 7467	. . 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 5905	. 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 33968	. 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18		zex 12545	. . . 4 ⊢ ℤ ∈ V
19	3	fvexi 6875	. . . . . 6 ⊢ 𝐿 ∈ V
20	19	cnvex 7904	. . . . 5 ⊢ ◡𝐿 ∈ V
21		imaexg 7892	. . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V)
22	20, 21	ax-mp 5	. . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V
23	18, 22	mpoex 8061	. . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
24	23	rnex 7889	. 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
25	16, 17, 24	fvmpt 6971	1 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598  ^◡ccnv 5640  ran crn 5642   “ cima 5644  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   / cdiv 11842  ℤcz 12536  1_rcur 20097  Unitcui 20271  /_rcdvr 20316  ℤRHomczrh 21416  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

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-ral 3046  df-rex 3055  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-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  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-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-neg 11415  df-z 12537  df-qqh 33968

This theorem is referenced by:  qqhval2  33979