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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibopelval3 | Structured version Visualization version GIF version |
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) |
Ref | Expression |
---|---|
dibval3.b | ⊢ 𝐵 = (Base‘𝐾) |
dibval3.l | ⊢ ≤ = (le‘𝐾) |
dibval3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibval3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dibval3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dibval3.o | ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dibval3.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibopelval3 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibval3.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dibval3.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dibval3.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dibval3.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | dibval3.o | . . 3 ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
6 | eqid 2725 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
7 | dibval3.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibopelval2 40748 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑆 = 0 ))) |
9 | dibval3.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
10 | 1, 2, 3, 4, 9, 6 | diaelval 40636 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))) |
11 | 10 | anbi1d 629 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑆 = 0 ) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 0 ))) |
12 | 8, 11 | bitrd 278 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 〈cop 4636 class class class wbr 5149 ↦ cmpt 5232 I cid 5575 ↾ cres 5680 ‘cfv 6549 Basecbs 17183 lecple 17243 LHypclh 39587 LTrncltrn 39704 trLctrl 39761 DIsoAcdia 40631 DIsoBcdib 40741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-disoa 40632 df-dib 40742 |
This theorem is referenced by: dihord2cN 40824 dihord11b 40825 dihopelvalbN 40841 dihopelvalcpre 40851 dihjatcclem4 41024 |
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