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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmtcomN | Structured version Visualization version GIF version |
Description: Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 31278 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cmtcom.b | ⊢ 𝐵 = (Base‘𝐾) |
cmtcom.c | ⊢ 𝐶 = (cm‘𝐾) |
Ref | Expression |
---|---|
cmtcomN | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmtcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cmtcom.c | . . 3 ⊢ 𝐶 = (cm‘𝐾) | |
3 | 1, 2 | cmtcomlemN 38582 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋)) |
4 | 1, 2 | cmtcomlemN 38582 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 → 𝑋𝐶𝑌)) |
5 | 4 | 3com23 1125 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 → 𝑋𝐶𝑌)) |
6 | 3, 5 | impbid 211 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 Basecbs 17151 cmccmtN 38507 OMLcoml 38509 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-proset 18258 df-poset 18276 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-lat 18395 df-oposet 38510 df-cmtN 38511 df-ol 38512 df-oml 38513 |
This theorem is referenced by: cmt3N 38585 cmtbr3N 38588 omlmod1i2N 38594 |
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