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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmt4N | Structured version Visualization version GIF version |
Description: Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31605 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 |
---|---|
cmt4N | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶( ⊥ ‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmt2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cmt2.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
3 | cmt2.c | . . 3 ⊢ 𝐶 = (cm‘𝐾) | |
4 | 1, 2, 3 | cmt2N 39193 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋𝐶( ⊥ ‘𝑌))) |
5 | omlop 39184 | . . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
6 | 5 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) |
7 | simp3 1136 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
8 | 1, 2 | opoccl 39137 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
9 | 6, 7, 8 | syl2anc 583 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
10 | 1, 2, 3 | cmt3N 39194 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋𝐶( ⊥ ‘𝑌) ↔ ( ⊥ ‘𝑋)𝐶( ⊥ ‘𝑌))) |
11 | 9, 10 | syld3an3 1407 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶( ⊥ ‘𝑌) ↔ ( ⊥ ‘𝑋)𝐶( ⊥ ‘𝑌))) |
12 | 4, 11 | bitrd 279 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶( ⊥ ‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 class class class wbr 5149 ‘cfv 6558 Basecbs 17234 occoc 17295 OPcops 39115 cmccmtN 39116 OMLcoml 39118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 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 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-proset 18341 df-poset 18359 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-lat 18478 df-oposet 39119 df-cmtN 39120 df-ol 39121 df-oml 39122 |
This theorem is referenced by: cmtbr2N 39196 omlfh3N 39202 |
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