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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 2821 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
7 | dibval3.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibopelval2 38275 | . 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 38163 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))) |
11 | 10 | anbi1d 631 | . 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 281 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ↦ cmpt 5138 I cid 5453 ↾ cres 5551 ‘cfv 6349 Basecbs 16477 lecple 16566 LHypclh 37114 LTrncltrn 37231 trLctrl 37288 DIsoAcdia 38158 DIsoBcdib 38268 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-disoa 38159 df-dib 38269 |
This theorem is referenced by: dihord2cN 38351 dihord11b 38352 dihopelvalbN 38368 dihopelvalcpre 38378 dihjatcclem4 38551 |
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