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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 2727 | . . . 4 ⊢ (𝑓 = 𝑅 → ℤ = ℤ) | |
2 | fveq2 6901 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅)) | |
3 | qqhval.3 | . . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
4 | 2, 3 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿) |
5 | 4 | cnveqd 5882 | . . . . 5 ⊢ (𝑓 = 𝑅 → ◡(ℤRHom‘𝑓) = ◡𝐿) |
6 | fveq2 6901 | . . . . 5 ⊢ (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅)) | |
7 | 5, 6 | imaeq12d 6070 | . . . 4 ⊢ (𝑓 = 𝑅 → (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) = (◡𝐿 “ (Unit‘𝑅))) |
8 | fveq2 6901 | . . . . . . 7 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = (/r‘𝑅)) | |
9 | qqhval.1 | . . . . . . 7 ⊢ / = (/r‘𝑅) | |
10 | 8, 9 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑓 = 𝑅 → (/r‘𝑓) = / ) |
11 | 4 | fveq1d 6903 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿‘𝑥)) |
12 | 4 | fveq1d 6903 | . . . . . 6 ⊢ (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿‘𝑦)) |
13 | 10, 11, 12 | oveq123d 7445 | . . . . 5 ⊢ (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿‘𝑥) / (𝐿‘𝑦))) |
14 | 13 | opeq2d 4886 | . . . 4 ⊢ (𝑓 = 𝑅 → 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉 = 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) |
15 | 1, 7, 14 | mpoeq123dv 7500 | . . 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 5944 | . 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 33788 | . 2 ⊢ ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ 〈(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r‘𝑓)((ℤRHom‘𝑓)‘𝑦))〉)) | |
18 | zex 12619 | . . . 4 ⊢ ℤ ∈ V | |
19 | 3 | fvexi 6915 | . . . . . 6 ⊢ 𝐿 ∈ V |
20 | 19 | cnvex 7938 | . . . . 5 ⊢ ◡𝐿 ∈ V |
21 | imaexg 7926 | . . . . 5 ⊢ (◡𝐿 ∈ V → (◡𝐿 “ (Unit‘𝑅)) ∈ V) | |
22 | 20, 21 | ax-mp 5 | . . . 4 ⊢ (◡𝐿 “ (Unit‘𝑅)) ∈ V |
23 | 18, 22 | mpoex 8093 | . . 3 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
24 | 23 | rnex 7923 | . 2 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ 〈(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))〉) ∈ V |
25 | 16, 17, 24 | fvmpt 7009 | 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 1534 ∈ wcel 2099 Vcvv 3462 〈cop 4639 ◡ccnv 5681 ran crn 5683 “ cima 5685 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 / cdiv 11921 ℤcz 12610 1rcur 20164 Unitcui 20337 /rcdvr 20382 ℤRHomczrh 21489 ℚHomcqqh 33787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-neg 11497 df-z 12611 df-qqh 33788 |
This theorem is referenced by: qqhval2 33797 |
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