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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opcon3b | Structured version Visualization version GIF version |
Description: Contraposition law for orthoposets. (chcon3i 31296 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
opoccl.b | ⊢ 𝐵 = (Base‘𝐾) |
opoccl.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opcon3b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6902 | . . 3 ⊢ (𝑌 = 𝑋 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)) | |
2 | 1 | eqcoms 2736 | . 2 ⊢ (𝑋 = 𝑌 → ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋)) |
3 | fveq2 6902 | . . . 4 ⊢ (( ⊥ ‘𝑋) = ( ⊥ ‘𝑌) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌))) | |
4 | 3 | eqcoms 2736 | . . 3 ⊢ (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌))) |
5 | opoccl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
6 | opoccl.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
7 | 5, 6 | opococ 38699 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
8 | 7 | 3adant3 1129 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
9 | 5, 6 | opococ 38699 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
10 | 9 | 3adant2 1128 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
11 | 8, 10 | eqeq12d 2744 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌)) ↔ 𝑋 = 𝑌)) |
12 | 4, 11 | imbitrid 243 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) = ( ⊥ ‘𝑋) → 𝑋 = 𝑌)) |
13 | 2, 12 | impbid2 225 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ( ⊥ ‘𝑌) = ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 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-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-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 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-ov 7429 df-oposet 38680 |
This theorem is referenced by: opcon2b 38701 omllaw4 38750 cmtbr2N 38757 cvrcmp2 38788 lhpmod2i2 39543 lhpmod6i1 39544 |
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