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Mirrors > Home > MPE Home > Th. List > Mathboxes > dicval2 | Structured version Visualization version GIF version |
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.) |
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) |
dicval2.g | ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) |
Ref | Expression |
---|---|
dicval2 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dicval.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | dicval.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | dicval.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dicval.p | . . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
5 | dicval.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | dicval.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
7 | dicval.i | . . 3 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dicval 38194 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)}) |
9 | dicval2.g | . . . . . 6 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) | |
10 | 9 | fveq2i 6667 | . . . . 5 ⊢ (𝑠‘𝐺) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) |
11 | 10 | eqeq2i 2834 | . . . 4 ⊢ (𝑓 = (𝑠‘𝐺) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))) |
12 | 11 | anbi1i 623 | . . 3 ⊢ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)) |
13 | 12 | opabbii 5125 | . 2 ⊢ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} |
14 | 8, 13 | syl6eqr 2874 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 {copab 5120 ‘cfv 6349 ℩crio 7102 lecple 16562 occoc 16563 Atomscatm 36281 LHypclh 37002 LTrncltrn 37119 TEndoctendo 37770 DIsoCcdic 38190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-dic 38191 |
This theorem is referenced by: dicelval3 38198 diclspsn 38212 dih1dimatlem 38347 |
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