| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dicelvalN | Structured version Visualization version GIF version | ||
| Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dicval.l | ⊢ ≤ = (le‘𝐾) |
| dicval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| dicval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| dicval.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
| dicval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| dicval.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
| dicval.i | ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) |
| Ref | Expression |
|---|---|
| dicelvalN | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dicval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 2 | dicval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 3 | dicval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 4 | dicval.p | . . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
| 5 | dicval.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 6 | dicval.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
| 7 | dicval.i | . . . 4 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | dicval 41178 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)}) |
| 9 | 8 | eleq2d 2827 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})) |
| 10 | vex 3484 | . . . . . 6 ⊢ 𝑓 ∈ V | |
| 11 | vex 3484 | . . . . . 6 ⊢ 𝑠 ∈ V | |
| 12 | 10, 11 | op1std 8024 | . . . . 5 ⊢ (𝑌 = 〈𝑓, 𝑠〉 → (1st ‘𝑌) = 𝑓) |
| 13 | 10, 11 | op2ndd 8025 | . . . . . 6 ⊢ (𝑌 = 〈𝑓, 𝑠〉 → (2nd ‘𝑌) = 𝑠) |
| 14 | 13 | fveq1d 6908 | . . . . 5 ⊢ (𝑌 = 〈𝑓, 𝑠〉 → ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))) |
| 15 | 12, 14 | eqeq12d 2753 | . . . 4 ⊢ (𝑌 = 〈𝑓, 𝑠〉 → ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))) |
| 16 | 13 | eleq1d 2826 | . . . 4 ⊢ (𝑌 = 〈𝑓, 𝑠〉 → ((2nd ‘𝑌) ∈ 𝐸 ↔ 𝑠 ∈ 𝐸)) |
| 17 | 15, 16 | anbi12d 632 | . . 3 ⊢ (𝑌 = 〈𝑓, 𝑠〉 → (((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸))) |
| 18 | 17 | elopaba 5818 | . 2 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸))) |
| 19 | 9, 18 | bitrdi 287 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸)))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 ‘cfv 6561 ℩crio 7387 1st c1st 8012 2nd c2nd 8013 lecple 17304 occoc 17305 Atomscatm 39264 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 DIsoCcdic 41174 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-1st 8014 df-2nd 8015 df-dic 41175 |
| This theorem is referenced by: dicelval2N 41184 |
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