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Mirrors > Home > MPE Home > Th. List > Mathboxes > dicelval1sta | Structured version Visualization version GIF version |
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.) |
Ref | Expression |
---|---|
dicelval1sta.l | ⊢ ≤ = (le‘𝐾) |
dicelval1sta.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dicelval1sta.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dicelval1sta.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
dicelval1sta.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dicelval1sta.i | ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dicelval1sta | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dicelval1sta.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
2 | dicelval1sta.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | dicelval1sta.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dicelval1sta.p | . . . . . 6 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
5 | dicelval1sta.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | eqid 2610 | . . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
7 | dicelval1sta.i | . . . . . 6 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dicval 35483 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) |
9 | 8 | eleq2d 2673 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})) |
10 | 9 | biimp3a 1424 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) |
11 | eqeq1 2614 | . . . . 5 ⊢ (𝑓 = (1st ‘𝑌) → (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ↔ (1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))) | |
12 | 11 | anbi1d 737 | . . . 4 ⊢ (𝑓 = (1st ‘𝑌) → ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)))) |
13 | fveq1 6102 | . . . . . 6 ⊢ (𝑠 = (2nd ‘𝑌) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))) | |
14 | 13 | eqeq2d 2620 | . . . . 5 ⊢ (𝑠 = (2nd ‘𝑌) → ((1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ↔ (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))) |
15 | eleq1 2676 | . . . . 5 ⊢ (𝑠 = (2nd ‘𝑌) → (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ↔ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))) | |
16 | 14, 15 | anbi12d 743 | . . . 4 ⊢ (𝑠 = (2nd ‘𝑌) → (((1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))) |
17 | 12, 16 | elopabi 7120 | . . 3 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} → ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))) |
18 | 10, 17 | syl 17 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))) |
19 | 18 | simpld 474 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 {copab 4642 ‘cfv 5804 ℩crio 6510 1st c1st 7057 2nd c2nd 7058 lecple 15775 occoc 15776 Atomscatm 33568 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 DIsoCcdic 35479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-1st 7059 df-2nd 7060 df-dic 35480 |
This theorem is referenced by: dicvaddcl 35497 dicvscacl 35498 |
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