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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 29363 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Cℋ cch 28708 ⊥cort 28709 ∨ℋ chj 28712 𝐶ℋ ccm 28715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-iota 6316 df-fv 6365 df-ov 7161 df-cm 29362 |
This theorem is referenced by: cmcmlem 29370 cmcm2i 29372 cmbr2i 29375 cmbr3i 29379 pjclem1 29974 pjci 29979 |
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