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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 12231 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 2899 | . . 3 ⊢ Ⅎ𝑦 ⊥ | |
2 | nfriota1 7389 | . . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓) | |
3 | 1, 2 | nffv 6912 | . 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) |
4 | riotaoc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
5 | riotaoc.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
6 | 4, 5 | opoccl 38698 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵) |
7 | 4, 5 | opoccl 38698 | . 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵) |
8 | riotaoc.a | . 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓)) | |
9 | fveq2 6902 | . 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | |
10 | 4, 5 | opoccl 38698 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵) |
11 | 4, 5 | opcon2b 38701 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥))) |
12 | 10, 11 | reuhypd 5423 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦)) |
13 | 3, 6, 7, 8, 9, 12 | riotaxfrd 7417 | 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 394 = wceq 1533 ∈ wcel 2098 ∃!wreu 3372 ‘cfv 6553 ℩crio 7381 Basecbs 17187 occoc 17248 OPcops 38676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-dm 5692 df-iota 6505 df-fv 6561 df-riota 7382 df-ov 7429 df-oposet 38680 |
This theorem is referenced by: glbconN 38881 glbconNOLD 38882 |
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