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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st2N | Structured version Visualization version GIF version |
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibelval1st2.b | ⊢ 𝐵 = (Base‘𝐾) |
dibelval1st2.l | ⊢ ≤ = (le‘𝐾) |
dibelval1st2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibelval1st2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dibelval1st2.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dibelval1st2.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibelval1st2N | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibelval1st2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dibelval1st2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dibelval1st2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2818 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
5 | dibelval1st2.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dibelval1st 38165 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) |
7 | dibelval1st2.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | dibelval1st2.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
9 | 1, 2, 3, 7, 8, 4 | diatrl 38060 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋) |
10 | 6, 9 | syld3an3 1401 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 1st c1st 7676 Basecbs 16471 lecple 16560 LHypclh 37000 LTrncltrn 37117 trLctrl 37174 DIsoAcdia 38044 DIsoBcdib 38154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1st 7678 df-disoa 38045 df-dib 38155 |
This theorem is referenced by: (None) |
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