Theorem qqhval 30564

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 2827	. . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ)
2		fveq2 6434	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3		qqhval.3	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅)
4	2, 3	syl6eqr 2880	. . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
5	4	cnveqd 5531	. . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿)
6		fveq2 6434	. . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
7	5, 6	imaeq12d 5709	. . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅)))
8		fveq2 6434	. . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅))
9		qqhval.1	. . . . . . 7 ⊢ / = (/r‘𝑅)
10	8, 9	syl6eqr 2880	. . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / )
11	4	fveq1d 6436	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥))
12	4	fveq1d 6436	. . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦))
13	10, 11, 12	oveq123d 6927	. . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦)))
14	13	opeq2d 4631	. . . 4 ⊢ (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
15	1, 7, 14	mpt2eq123dv 6978	. . 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 5586	. 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 30563	. 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18		zex 11714	. . . 4 ⊢ ℤ ∈ V
19	3	fvexi 6448	. . . . . 6 ⊢ 𝐿 ∈ V
20	19	cnvex 7376	. . . . 5 ⊢ ◡𝐿 ∈ V
21		imaexg 7366	. . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V)
22	20, 21	ax-mp 5	. . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V
23	18, 22	mpt2ex 7511	. . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
24	23	rnex 7363	. 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) ∈ V
25	16, 17, 24	fvmpt 6530	1 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3415  ⟨cop 4404  ^◡ccnv 5342  ran crn 5344   “ cima 5346  ‘cfv 6124  (class class class)co 6906   ↦ cmpt2 6908   / cdiv 11010  ℤcz 11705  1_rcur 18856  Unitcui 18994  /_rcdvr 19037  ℤRHomczrh 20209  ℚHomcqqh 30562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-neg 10589  df-z 11706  df-qqh 30563

This theorem is referenced by:  qqhval2  30572