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Mirrors > Home > MPE Home > Th. List > Mathboxes > dicopelval2 | Structured version Visualization version GIF version |
Description: Membership in value of 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) |
dicelval2.f | ⊢ 𝐹 ∈ V |
dicelval2.s | ⊢ 𝑆 ∈ V |
Ref | Expression |
---|---|
dicopelval2 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑄) ↔ (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 | dicelval2.f | . . 3 ⊢ 𝐹 ∈ V | |
9 | dicelval2.s | . . 3 ⊢ 𝑆 ∈ V | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dicopelval 38307 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑆 ∈ 𝐸))) |
11 | dicval2.g | . . . . 5 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) | |
12 | 11 | fveq2i 6667 | . . . 4 ⊢ (𝑆‘𝐺) = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) |
13 | 12 | eqeq2i 2834 | . . 3 ⊢ (𝐹 = (𝑆‘𝐺) ↔ 𝐹 = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))) |
14 | 13 | anbi1i 625 | . 2 ⊢ ((𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸) ↔ (𝐹 = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑆 ∈ 𝐸)) |
15 | 10, 14 | syl6bbr 291 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4566 class class class wbr 5058 ‘cfv 6349 ℩crio 7107 lecple 16566 occoc 16567 Atomscatm 36393 LHypclh 37114 LTrncltrn 37231 TEndoctendo 37882 DIsoCcdic 38302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 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 7108 df-dic 38303 |
This theorem is referenced by: diclspsn 38324 cdlemn11a 38337 dihopelvalcqat 38376 dihopelvalcpre 38378 dihord6apre 38386 |
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