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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 31612 | . 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 1536 ∈ wcel 2105 ∩ cin 3961 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Cℋ cch 30957 ⊥cort 30958 ∨ℋ chj 30961 𝐶ℋ ccm 30964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-iota 6515 df-fv 6570 df-ov 7433 df-cm 31611 |
This theorem is referenced by: cmcmlem 31619 cmcm2i 31621 cmbr2i 31624 cmbr3i 31628 pjclem1 32223 pjci 32228 |
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