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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st1 | 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.) |
Ref | Expression |
---|---|
dibelval1st1.b | ⊢ 𝐵 = (Base‘𝐾) |
dibelval1st1.l | ⊢ ≤ = (le‘𝐾) |
dibelval1st1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibelval1st1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dibelval1st1.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibelval1st1 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibelval1st1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dibelval1st1.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dibelval1st1.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2819 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
5 | dibelval1st1.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dibelval1st 38277 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) |
7 | dibelval1st1.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 7, 4 | diael 38171 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (1st ‘𝑌) ∈ 𝑇) |
9 | 6, 8 | syld3an3 1404 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 1st c1st 7679 Basecbs 16475 lecple 16564 LHypclh 37112 LTrncltrn 37229 DIsoAcdia 38156 DIsoBcdib 38266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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 7681 df-disoa 38157 df-dib 38267 |
This theorem is referenced by: diblss 38298 |
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