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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 31343 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 38624 | . . 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 38623 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) |
7 | omlop 38615 | . . . . 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 38568 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
11 | 8, 9, 10 | syl2anc 583 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 1, 3 | cmtcomN 38623 | . . 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 311 | 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 5139 ‘cfv 6534 Basecbs 17149 occoc 17210 OPcops 38546 cmccmtN 38547 OMLcoml 38549 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-lat 18393 df-oposet 38550 df-cmtN 38551 df-ol 38552 df-oml 38553 |
This theorem is referenced by: cmt4N 38626 omlspjN 38635 |
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