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Mathbox for Norm Megill |
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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 2731 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
5 | dibelval1st2.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dibelval1st 39825 | . 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 39720 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋) |
10 | 6, 9 | syld3an3 1409 | 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 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ‘cfv 6532 1st c1st 7955 Basecbs 17126 lecple 17186 LHypclh 38660 LTrncltrn 38777 trLctrl 38834 DIsoAcdia 39704 DIsoBcdib 39814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-1st 7957 df-disoa 39705 df-dib 39815 |
This theorem is referenced by: (None) |
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