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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 2736 | . . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ) | |
2 | fveq2 6907 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅)) | |
3 | qqhval.3 | . . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿) |
5 | 4 | cnveqd 5889 | . . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿) |
6 | fveq2 6907 | . . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅)) | |
7 | 5, 6 | imaeq12d 6081 | . . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅))) |
8 | fveq2 6907 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅)) | |
9 | qqhval.1 | . . . . . . 7 ⊢ / = (/r‘𝑅) | |
10 | 8, 9 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / ) |
11 | 4 | fveq1d 6909 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥)) |
12 | 4 | fveq1d 6909 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦)) |
13 | 10, 11, 12 | oveq123d 7452 | . . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦))) |
14 | 13 | opeq2d 4885 | . . . 4 ⊢ (𝑓 = 𝑅 → 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉 = 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) |
15 | 1, 7, 14 | mpoeq123dv 7508 | . . 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 5952 | . 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 33934 | . 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉)) | |
18 | zex 12620 | . . . 4 ⊢ ℤ ∈ V | |
19 | 3 | fvexi 6921 | . . . . . 6 ⊢ 𝐿 ∈ V |
20 | 19 | cnvex 7948 | . . . . 5 ⊢ ◡𝐿 ∈ V |
21 | imaexg 7936 | . . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V) | |
22 | 20, 21 | ax-mp 5 | . . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V |
23 | 18, 22 | mpoex 8103 | . . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
24 | 23 | rnex 7933 | . 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
25 | 16, 17, 24 | fvmpt 7016 | 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 2106 Vcvv 3478 〈cop 4637 ◡ccnv 5688 ran crn 5690 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 / cdiv 11918 ℤcz 12611 1rcur 20199 Unitcui 20372 /rcdvr 20417 ℤRHomczrh 21528 ℚHomcqqh 33933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-neg 11493 df-z 12612 df-qqh 33934 |
This theorem is referenced by: qqhval2 33945 |
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