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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 31603 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Cℋ cch 30948 ⊥cort 30949 ∨ℋ chj 30952 𝐶ℋ ccm 30955 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 df-ov 7434 df-cm 31602 |
| This theorem is referenced by: cmcmlem 31610 cmcm2i 31612 cmbr2i 31615 cmbr3i 31619 pjclem1 32214 pjci 32219 |
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