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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 2825 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dihvalb 37307 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋)) |
7 | 6 | eleq2d 2892 | . 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 37218 | . 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 271 | 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 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 〈cop 4405 class class class wbr 4875 ↦ cmpt 4954 I cid 5251 ↾ cres 5348 ‘cfv 6127 Basecbs 16229 lecple 16319 LHypclh 36054 LTrncltrn 36171 trLctrl 36228 DIsoBcdib 37208 DIsoHcdih 37298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-disoa 37099 df-dib 37209 df-dih 37299 |
This theorem is referenced by: dihmeetlem1N 37360 dihglblem5apreN 37361 dihmeetlem4preN 37376 |
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