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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 2737 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
5 | dibelval1st2.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dibelval1st 38900 | . 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 38795 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋) |
10 | 6, 9 | syld3an3 1411 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 1st c1st 7759 Basecbs 16760 lecple 16809 LHypclh 37735 LTrncltrn 37852 trLctrl 37909 DIsoAcdia 38779 DIsoBcdib 38889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-1st 7761 df-disoa 38780 df-dib 38890 |
This theorem is referenced by: (None) |
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