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 2822 | . . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ) | |
2 | fveq2 6670 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅)) | |
3 | qqhval.3 | . . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
4 | 2, 3 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿) |
5 | 4 | cnveqd 5746 | . . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿) |
6 | fveq2 6670 | . . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅)) | |
7 | 5, 6 | imaeq12d 5930 | . . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅))) |
8 | fveq2 6670 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅)) | |
9 | qqhval.1 | . . . . . . 7 ⊢ / = (/r‘𝑅) | |
10 | 8, 9 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / ) |
11 | 4 | fveq1d 6672 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥)) |
12 | 4 | fveq1d 6672 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦)) |
13 | 10, 11, 12 | oveq123d 7177 | . . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦))) |
14 | 13 | opeq2d 4810 | . . . 4 ⊢ (𝑓 = 𝑅 → 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉 = 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) |
15 | 1, 7, 14 | mpoeq123dv 7229 | . . 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 5808 | . 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 31214 | . 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉)) | |
18 | zex 11991 | . . . 4 ⊢ ℤ ∈ V | |
19 | 3 | fvexi 6684 | . . . . . 6 ⊢ 𝐿 ∈ V |
20 | 19 | cnvex 7630 | . . . . 5 ⊢ ◡𝐿 ∈ V |
21 | imaexg 7620 | . . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V) | |
22 | 20, 21 | ax-mp 5 | . . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V |
23 | 18, 22 | mpoex 7777 | . . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
24 | 23 | rnex 7617 | . 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
25 | 16, 17, 24 | fvmpt 6768 | 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 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ◡ccnv 5554 ran crn 5556 “ cima 5558 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 / cdiv 11297 ℤcz 11982 1rcur 19251 Unitcui 19389 /rcdvr 19432 ℤRHomczrh 20647 ℚHomcqqh 31213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-neg 10873 df-z 11983 df-qqh 31214 |
This theorem is referenced by: qqhval2 31223 |
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