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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmt3N | Structured version Visualization version GIF version |
Description: Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31477 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cmt2.b | ⊢ 𝐵 = (Base‘𝐾) |
cmt2.o | ⊢ ⊥ = (oc‘𝐾) |
cmt2.c | ⊢ 𝐶 = (cm‘𝐾) |
Ref | Expression |
---|---|
cmt3N | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmt2.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cmt2.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
3 | cmt2.c | . . . 4 ⊢ 𝐶 = (cm‘𝐾) | |
4 | 1, 2, 3 | cmt2N 38852 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
5 | 4 | 3com23 1123 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
6 | 1, 3 | cmtcomN 38851 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) |
7 | omlop 38843 | . . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
8 | 7 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) |
9 | simp2 1134 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | 1, 2 | opoccl 38796 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
11 | 8, 9, 10 | syl2anc 582 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 1, 3 | cmtcomN 38851 | . . 3 ⊢ ((𝐾 ∈ OML ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋)𝐶𝑌 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
13 | 11, 12 | syld3an2 1408 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋)𝐶𝑌 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
14 | 5, 6, 13 | 3bitr4d 310 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 Basecbs 17183 occoc 17244 OPcops 38774 cmccmtN 38775 OMLcoml 38777 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-proset 18290 df-poset 18308 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-lat 18427 df-oposet 38778 df-cmtN 38779 df-ol 38780 df-oml 38781 |
This theorem is referenced by: cmt4N 38854 omlspjN 38863 |
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