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Theorem oldmm1 38598
Description: De Morgan's law for meet in an ortholattice. (chdmm1 31283 analog.) (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
oldmm1.b 𝐡 = (Baseβ€˜πΎ)
oldmm1.j ∨ = (joinβ€˜πΎ)
oldmm1.m ∧ = (meetβ€˜πΎ)
oldmm1.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
oldmm1 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
Proof of Theorem oldmm1
StepHypRef Expression
1 oldmm1.b . 2 𝐡 = (Baseβ€˜πΎ)
2 eqid 2726 . 2 (leβ€˜πΎ) = (leβ€˜πΎ)
3 ollat 38594 . . 3 (𝐾 ∈ OL β†’ 𝐾 ∈ Lat)
433ad2ant1 1130 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
5 olop 38595 . . . 4 (𝐾 ∈ OL β†’ 𝐾 ∈ OP)
653ad2ant1 1130 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
7 oldmm1.m . . . . 5 ∧ = (meetβ€˜πΎ)
81, 7latmcl 18403 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
93, 8syl3an1 1160 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
10 oldmm1.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
111, 10opoccl 38575 . . 3 ((𝐾 ∈ OP ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∈ 𝐡)
126, 9, 11syl2anc 583 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∈ 𝐡)
131, 10opoccl 38575 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
145, 13sylan 579 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
15143adant3 1129 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
161, 10opoccl 38575 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
175, 16sylan 579 . . . 4 ((𝐾 ∈ OL ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
18173adant2 1128 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
19 oldmm1.j . . . 4 ∨ = (joinβ€˜πΎ)
201, 19latjcl 18402 . . 3 ((𝐾 ∈ Lat ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
214, 15, 18, 20syl3anc 1368 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
221, 2, 19latlej1 18411 . . . . . 6 ((𝐾 ∈ Lat ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
234, 15, 18, 22syl3anc 1368 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
24 simp2 1134 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
251, 2, 10oplecon1b 38582 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)𝑋))
266, 24, 21, 25syl3anc 1368 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)𝑋))
2723, 26mpbid 231 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)𝑋)
281, 2, 19latlej2 18412 . . . . . 6 ((𝐾 ∈ Lat ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
294, 15, 18, 28syl3anc 1368 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
30 simp3 1135 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
311, 2, 10oplecon1b 38582 . . . . . 6 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡 ∧ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)π‘Œ))
326, 30, 21, 31syl3anc 1368 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)π‘Œ))
3329, 32mpbid 231 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)π‘Œ)
341, 10opoccl 38575 . . . . . 6 ((𝐾 ∈ OP ∧ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
356, 21, 34syl2anc 583 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
361, 2, 7latlem12 18429 . . . . 5 ((𝐾 ∈ Lat ∧ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)𝑋 ∧ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)π‘Œ) ↔ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)(𝑋 ∧ π‘Œ)))
374, 35, 24, 30, 36syl13anc 1369 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)𝑋 ∧ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)π‘Œ) ↔ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)(𝑋 ∧ π‘Œ)))
3827, 33, 37mpbi2and 709 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)(𝑋 ∧ π‘Œ))
391, 2, 10oplecon1b 38582 . . . 4 ((𝐾 ∈ OP ∧ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)(𝑋 ∧ π‘Œ) ↔ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ))(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))))
406, 21, 9, 39syl3anc 1368 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))(leβ€˜πΎ)(𝑋 ∧ π‘Œ) ↔ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ))(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))))
4138, 40mpbid 231 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ))(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
421, 2, 7latmle1 18427 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋)
433, 42syl3an1 1160 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋)
441, 2, 10oplecon3b 38581 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋 ↔ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))))
456, 9, 24, 44syl3anc 1368 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋 ↔ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))))
4643, 45mpbid 231 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ)))
471, 2, 7latmle2 18428 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ)
483, 47syl3an1 1160 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ)
491, 2, 10oplecon3b 38581 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ ↔ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))))
506, 9, 30, 49syl3anc 1368 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ ↔ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))))
5148, 50mpbid 231 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ)))
521, 2, 19latjle12 18413 . . . 4 ((𝐾 ∈ Lat ∧ (( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡 ∧ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∈ 𝐡)) β†’ ((( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∧ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))) ↔ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))))
534, 15, 18, 12, 52syl13anc 1369 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∧ ( βŠ₯ β€˜π‘Œ)(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))) ↔ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ))))
5446, 51, 53mpbi2and 709 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))(leβ€˜πΎ)( βŠ₯ β€˜(𝑋 ∧ π‘Œ)))
551, 2, 4, 12, 21, 41, 54latasymd 18408 1 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  occoc 17212  joincjn 18274  meetcmee 18275  Latclat 18394  OPcops 38553  OLcol 38555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18258  df-poset 18276  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-lat 18395  df-oposet 38557  df-ol 38559
This theorem is referenced by:  oldmm2  38599  oldmm3N  38600  cmtcomlemN  38629  cmtbr2N  38634  omlfh1N  38639  cvrexch  38802  lhpmod2i2  39420  lhpmod6i1  39421  doca2N  40508  djajN  40519
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