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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 29007 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 35324 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋)) |
4 | 1, 2 | cmtcomlemN 35324 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 → 𝑋𝐶𝑌)) |
5 | 4 | 3com23 1162 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 → 𝑋𝐶𝑌)) |
6 | 3, 5 | impbid 204 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ‘cfv 6124 Basecbs 16223 cmccmtN 35249 OMLcoml 35251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-proset 17282 df-poset 17300 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-lat 17400 df-oposet 35252 df-cmtN 35253 df-ol 35254 df-oml 35255 |
This theorem is referenced by: cmt3N 35327 cmtbr3N 35330 omlmod1i2N 35336 |
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