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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 11002 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 2764 | . . 3 ⊢ Ⅎ𝑦 ⊥ | |
2 | nfriota1 6618 | . . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓) | |
3 | 1, 2 | nffv 6198 | . 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) |
4 | riotaoc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
5 | riotaoc.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
6 | 4, 5 | opoccl 34481 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵) |
7 | 4, 5 | opoccl 34481 | . 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵) |
8 | riotaoc.a | . 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓)) | |
9 | fveq2 6191 | . 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) | |
10 | 4, 5 | opoccl 34481 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵) |
11 | 4, 5 | opcon2b 34484 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥))) |
12 | 10, 11 | reuhypd 4895 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦)) |
13 | 3, 6, 7, 8, 9, 12 | riotaxfrd 6642 | 1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃!wreu 2914 ‘cfv 5888 ℩crio 6610 Basecbs 15857 occoc 15949 OPcops 34459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-riota 6611 df-ov 6653 df-oposet 34463 |
This theorem is referenced by: glbconN 34663 |
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