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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 12135 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 2908 | . . 3 ⊢ Ⅎ𝑦 ⊥ | |
2 | nfriota1 7321 | . . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓) | |
3 | 1, 2 | nffv 6853 | . 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) |
4 | riotaoc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
5 | riotaoc.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
6 | 4, 5 | opoccl 37659 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵) |
7 | 4, 5 | opoccl 37659 | . 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵) |
8 | riotaoc.a | . 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓)) | |
9 | fveq2 6843 | . 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | |
10 | 4, 5 | opoccl 37659 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵) |
11 | 4, 5 | opcon2b 37662 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥))) |
12 | 10, 11 | reuhypd 5375 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦)) |
13 | 3, 6, 7, 8, 9, 12 | riotaxfrd 7349 | 1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃!wreu 3352 ‘cfv 6497 ℩crio 7313 Basecbs 17084 occoc 17142 OPcops 37637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-dm 5644 df-iota 6449 df-fv 6505 df-riota 7314 df-ov 7361 df-oposet 37641 |
This theorem is referenced by: glbconN 37842 glbconNOLD 37843 |
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