Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qqhval | Structured version Visualization version GIF version |
Description: Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
qqhval.1 | ⊢ / = (/r‘𝑅) |
qqhval.2 | ⊢ 1 = (1r‘𝑅) |
qqhval.3 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
qqhval | ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2611 | . . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ) | |
2 | fveq2 6103 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅)) | |
3 | qqhval.3 | . . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
4 | 2, 3 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿) |
5 | 4 | cnveqd 5220 | . . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿) |
6 | fveq2 6103 | . . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅)) | |
7 | 5, 6 | imaeq12d 5386 | . . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅))) |
8 | fveq2 6103 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅)) | |
9 | qqhval.1 | . . . . . . 7 ⊢ / = (/r‘𝑅) | |
10 | 8, 9 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / ) |
11 | 4 | fveq1d 6105 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥)) |
12 | 4 | fveq1d 6105 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦)) |
13 | 10, 11, 12 | oveq123d 6570 | . . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦))) |
14 | 13 | opeq2d 4347 | . . . 4 ⊢ (𝑓 = 𝑅 → 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉 = 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) |
15 | 1, 7, 14 | mpt2eq123dv 6615 | . . 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 5274 | . 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 29345 | . 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉)) | |
18 | zex 11263 | . . . 4 ⊢ ℤ ∈ V | |
19 | fvex 6113 | . . . . . . 7 ⊢ (ℤRHom‘𝑅) ∈ V | |
20 | 3, 19 | eqeltri 2684 | . . . . . 6 ⊢ 𝐿 ∈ V |
21 | 20 | cnvex 7006 | . . . . 5 ⊢ ◡𝐿 ∈ V |
22 | imaexg 6995 | . . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V) | |
23 | 21, 22 | ax-mp 5 | . . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V |
24 | 18, 23 | mpt2ex 7136 | . . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
25 | 24 | rnex 6992 | . 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
26 | 16, 17, 25 | fvmpt 6191 | 1 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ◡ccnv 5037 ran crn 5039 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 / cdiv 10563 ℤcz 11254 1rcur 18324 Unitcui 18462 /rcdvr 18505 ℤRHomczrh 19667 ℚHomcqqh 29344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-neg 10148 df-z 11255 df-qqh 29345 |
This theorem is referenced by: qqhval2 29354 |
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