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Mirrors > Home > HSE Home > Th. List > cmbri | Structured version Visualization version GIF version |
Description: Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjoml2.1 | ⊢ 𝐴 ∈ Cℋ |
pjoml2.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
cmbri | ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjoml2.1 | . 2 ⊢ 𝐴 ∈ Cℋ | |
2 | pjoml2.2 | . 2 ⊢ 𝐵 ∈ Cℋ | |
3 | cmbr 31306 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 Cℋ cch 30651 ⊥cort 30652 ∨ℋ chj 30655 𝐶ℋ ccm 30658 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-iota 6485 df-fv 6541 df-ov 7404 df-cm 31305 |
This theorem is referenced by: cmcmlem 31313 cmcm2i 31315 cmbr2i 31318 cmbr3i 31322 pjclem1 31917 pjci 31922 |
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