Theorem qqhval 34156

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 2740	. . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ)
2		fveq2 6827	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3		qqhval.3	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅)
4	2, 3	eqtr4di 2792	. . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
5	4	cnveqd 5817	. . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿)
6		fveq2 6827	. . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
7	5, 6	imaeq12d 6013	. . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅)))
8		fveq2 6827	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅))
9		qqhval.1	. . . . . . 7 ⊢ / = (/r‘𝑅)
10	8, 9	eqtr4di 2792	. . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / )
11	4	fveq1d 6829	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥))
12	4	fveq1d 6829	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦))
13	10, 11, 12	oveq123d 7377	. . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦)))
14	13	opeq2d 4811	. . . 4 ⊢ (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
15	1, 7, 14	mpoeq123dv 7431	. . 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 5880	. 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 34155	. 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18		zex 12524	. . . 4 ⊢ ℤ ∈ V
19	3	fvexi 6841	. . . . . 6 ⊢ 𝐿 ∈ V
20	19	cnvex 7865	. . . . 5 ⊢ ◡𝐿 ∈ V
21		imaexg 7853	. . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V)
22	20, 21	ax-mp 5	. . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V
23	18, 22	mpoex 8021	. . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
24	23	rnex 7850	. 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
25	16, 17, 24	fvmpt 6935	1 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561  ^◡ccnv 5617  ran crn 5619   “ cima 5621  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358   / cdiv 11798  ℤcz 12515  1_rcur 20153  Unitcui 20326  /_rcdvr 20371  ℤRHomczrh 21474  ℚHomcqqh 34154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-neg 11371  df-z 12516  df-qqh 34155

This theorem is referenced by:  qqhval2  34166