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Mirrors > Home > MPE Home > Th. List > Mathboxes > cmtidN | Structured version Visualization version GIF version |
Description: Any element commutes with itself. (cmidi 29389 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cmtid.b | ⊢ 𝐵 = (Base‘𝐾) |
cmtid.c | ⊢ 𝐶 = (cm‘𝐾) |
Ref | Expression |
---|---|
cmtidN | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → 𝑋𝐶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omllat 36380 | . . 3 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
2 | cmtid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | eqid 2823 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | 2, 3 | latref 17665 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
5 | 1, 4 | sylan 582 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
6 | cmtid.c | . . . 4 ⊢ 𝐶 = (cm‘𝐾) | |
7 | 2, 3, 6 | lecmtN 36394 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋(le‘𝐾)𝑋 → 𝑋𝐶𝑋)) |
8 | 7 | 3anidm23 1417 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → (𝑋(le‘𝐾)𝑋 → 𝑋𝐶𝑋)) |
9 | 5, 8 | mpd 15 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → 𝑋𝐶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 Latclat 17657 cmccmtN 36311 OMLcoml 36313 |
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-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-lat 17658 df-oposet 36314 df-cmtN 36315 df-ol 36316 df-oml 36317 |
This theorem is referenced by: omlspjN 36399 |
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