| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cmt3N | Structured version Visualization version GIF version | ||
| Description: Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31604 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 39229 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
| 5 | 4 | 3com23 1127 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
| 6 | 1, 3 | cmtcomN 39228 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) |
| 7 | omlop 39220 | . . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
| 8 | 7 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) |
| 9 | simp2 1138 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 10 | 1, 2 | opoccl 39173 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
| 11 | 8, 9, 10 | syl2anc 584 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
| 12 | 1, 3 | cmtcomN 39228 | . . 3 ⊢ ((𝐾 ∈ OML ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋)𝐶𝑌 ↔ 𝑌𝐶( ⊥ ‘𝑋))) |
| 13 | 11, 12 | syld3an2 1413 | . 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5141 ‘cfv 6559 Basecbs 17243 occoc 17301 OPcops 39151 cmccmtN 39152 OMLcoml 39154 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-proset 18336 df-poset 18355 df-lub 18387 df-glb 18388 df-join 18389 df-meet 18390 df-lat 18473 df-oposet 39155 df-cmtN 39156 df-ol 39157 df-oml 39158 |
| This theorem is referenced by: cmt4N 39231 omlspjN 39240 |
| Copyright terms: Public domain | W3C validator |