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Theorem oldmm1 38087
Description: De Morgan's law for meet in an ortholattice. (chdmm1 30778 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 2733 . 2 (leβ€˜πΎ) = (leβ€˜πΎ)
3 ollat 38083 . . 3 (𝐾 ∈ OL β†’ 𝐾 ∈ Lat)
433ad2ant1 1134 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
5 olop 38084 . . . 4 (𝐾 ∈ OL β†’ 𝐾 ∈ OP)
653ad2ant1 1134 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
7 oldmm1.m . . . . 5 ∧ = (meetβ€˜πΎ)
81, 7latmcl 18393 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
93, 8syl3an1 1164 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
10 oldmm1.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
111, 10opoccl 38064 . . 3 ((𝐾 ∈ OP ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∈ 𝐡)
126, 9, 11syl2anc 585 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ π‘Œ)) ∈ 𝐡)
131, 10opoccl 38064 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
145, 13sylan 581 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
15143adant3 1133 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
161, 10opoccl 38064 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
175, 16sylan 581 . . . 4 ((𝐾 ∈ OL ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
18173adant2 1132 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
19 oldmm1.j . . . 4 ∨ = (joinβ€˜πΎ)
201, 19latjcl 18392 . . 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 1372 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
221, 2, 19latlej1 18401 . . . . . 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 1372 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
24 simp2 1138 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
251, 2, 10oplecon1b 38071 . . . . . 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 1372 . . . . 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 18402 . . . . . 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 1372 . . . . 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 1139 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
311, 2, 10oplecon1b 38071 . . . . . 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 1372 . . . . 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 38064 . . . . . 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 585 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
361, 2, 7latlem12 18419 . . . . 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 1373 . . . 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 711 . . 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 38071 . . . 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 1372 . . 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 18417 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋)
433, 42syl3an1 1164 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋)
441, 2, 10oplecon3b 38070 . . . . 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 1372 . . . 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 18418 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ)
483, 47syl3an1 1164 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ)
491, 2, 10oplecon3b 38070 . . . . 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 1372 . . . 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 18403 . . . 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 1373 . . 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 711 . 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 18398 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  occoc 17205  joincjn 18264  meetcmee 18265  Latclat 18384  OPcops 38042  OLcol 38044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-lat 18385  df-oposet 38046  df-ol 38048
This theorem is referenced by:  oldmm2  38088  oldmm3N  38089  cmtcomlemN  38118  cmtbr2N  38123  omlfh1N  38128  cvrexch  38291  lhpmod2i2  38909  lhpmod6i1  38910  doca2N  39997  djajN  40008
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