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Theorem oldmm1 38689
Description: De Morgan's law for meet in an ortholattice. (chdmm1 31348 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 2728 . 2 (leβ€˜πΎ) = (leβ€˜πΎ)
3 ollat 38685 . . 3 (𝐾 ∈ OL β†’ 𝐾 ∈ Lat)
433ad2ant1 1131 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
5 olop 38686 . . . 4 (𝐾 ∈ OL β†’ 𝐾 ∈ OP)
653ad2ant1 1131 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
7 oldmm1.m . . . . 5 ∧ = (meetβ€˜πΎ)
81, 7latmcl 18432 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
93, 8syl3an1 1161 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
10 oldmm1.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
111, 10opoccl 38666 . . 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 38666 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
145, 13sylan 579 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
15143adant3 1130 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
161, 10opoccl 38666 . . . . 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 1129 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
19 oldmm1.j . . . 4 ∨ = (joinβ€˜πΎ)
201, 19latjcl 18431 . . 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 1369 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
221, 2, 19latlej1 18440 . . . . . 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 1369 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹)(leβ€˜πΎ)(( βŠ₯ β€˜π‘‹) ∨ ( βŠ₯ β€˜π‘Œ)))
24 simp2 1135 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
251, 2, 10oplecon1b 38673 . . . . . 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 1369 . . . . 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 18441 . . . . . 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 1369 . . . . 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 1136 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
311, 2, 10oplecon1b 38673 . . . . . 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 1369 . . . . 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 38666 . . . . . 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 18458 . . . . 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 1370 . . . 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 38673 . . . 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 1369 . . 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 18456 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋)
433, 42syl3an1 1161 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)𝑋)
441, 2, 10oplecon3b 38672 . . . . 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 1369 . . . 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 18457 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ)
483, 47syl3an1 1161 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ)(leβ€˜πΎ)π‘Œ)
491, 2, 10oplecon3b 38672 . . . . 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 1369 . . . 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 18442 . . . 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 1370 . . 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 18437 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 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5148  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  occoc 17241  joincjn 18303  meetcmee 18304  Latclat 18423  OPcops 38644  OLcol 38646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-proset 18287  df-poset 18305  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-lat 18424  df-oposet 38648  df-ol 38650
This theorem is referenced by:  oldmm2  38690  oldmm3N  38691  cmtcomlemN  38720  cmtbr2N  38725  omlfh1N  38730  cvrexch  38893  lhpmod2i2  39511  lhpmod6i1  39512  doca2N  40599  djajN  40610
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