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Mathbox for Norm Megill |
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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 12245 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 2903 | . . 3 ⊢ Ⅎ𝑦 ⊥ | |
2 | nfriota1 7395 | . . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓) | |
3 | 1, 2 | nffv 6917 | . 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) |
4 | riotaoc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
5 | riotaoc.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
6 | 4, 5 | opoccl 39176 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵) |
7 | 4, 5 | opoccl 39176 | . 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵) |
8 | riotaoc.a | . 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓)) | |
9 | fveq2 6907 | . 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | |
10 | 4, 5 | opoccl 39176 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵) |
11 | 4, 5 | opcon2b 39179 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥))) |
12 | 10, 11 | reuhypd 5425 | . 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 1537 ∈ wcel 2106 ∃!wreu 3376 ‘cfv 6563 ℩crio 7387 Basecbs 17245 occoc 17306 OPcops 39154 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 df-riota 7388 df-ov 7434 df-oposet 39158 |
This theorem is referenced by: glbconN 39359 glbconNOLD 39360 |
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