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| Mirrors > Home > MPE Home > Th. List > Mathboxes > riotaocN | Structured version Visualization version GIF version | ||
| Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12247 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| riotaoc.b | ⊢ 𝐵 = (Base‘𝐾) | 
| riotaoc.o | ⊢ ⊥ = (oc‘𝐾) | 
| riotaoc.a | ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| riotaocN | ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑦 ⊥ | |
| 2 | nfriota1 7395 | . . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓) | |
| 3 | 1, 2 | nffv 6916 | . 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) | 
| 4 | riotaoc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 5 | riotaoc.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
| 6 | 4, 5 | opoccl 39195 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵) | 
| 7 | 4, 5 | opoccl 39195 | . 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵) | 
| 8 | riotaoc.a | . 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓)) | |
| 9 | fveq2 6906 | . 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | |
| 10 | 4, 5 | opoccl 39195 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵) | 
| 11 | 4, 5 | opcon2b 39198 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥))) | 
| 12 | 10, 11 | reuhypd 5419 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦)) | 
| 13 | 3, 6, 7, 8, 9, 12 | riotaxfrd 7422 | 1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!wreu 3378 ‘cfv 6561 ℩crio 7387 Basecbs 17247 occoc 17305 OPcops 39173 | 
| 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-nul 5306 | 
| 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-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-riota 7388 df-ov 7434 df-oposet 39177 | 
| This theorem is referenced by: glbconN 39378 glbconNOLD 39379 | 
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