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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihopelvalbN | Structured version Visualization version GIF version |
Description: Ordered pair member of the partial isomorphism H for argument under 𝑊. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihval3.b | ⊢ 𝐵 = (Base‘𝐾) |
dihval3.l | ⊢ ≤ = (le‘𝐾) |
dihval3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihval3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihval3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dihval3.o | ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dihval3.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihopelvalbN | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihval3.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihval3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | dihval3.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dihval3.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
5 | eqid 2734 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dihvalb 41219 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋)) |
7 | 6 | eleq2d 2824 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ 〈𝐹, 𝑆〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋))) |
8 | dihval3.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
9 | dihval3.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
10 | dihval3.o | . . 3 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
11 | 1, 2, 3, 8, 9, 10, 5 | dibopelval3 41130 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂))) |
12 | 7, 11 | bitrd 279 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 〈cop 4636 class class class wbr 5147 ↦ cmpt 5230 I cid 5581 ↾ cres 5690 ‘cfv 6562 Basecbs 17244 lecple 17304 LHypclh 39966 LTrncltrn 40083 trLctrl 40140 DIsoBcdib 41120 DIsoHcdih 41210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-disoa 41011 df-dib 41121 df-dih 41211 |
This theorem is referenced by: dihmeetlem1N 41272 dihglblem5apreN 41273 dihmeetlem4preN 41288 |
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