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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 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))〉)) |
Colors of variables: wff setvar class |
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 1rcur 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 |
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