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Theorem cmtbr3N 38427
Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 31116 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtbr2.b 𝐡 = (Baseβ€˜πΎ)
cmtbr2.j ∨ = (joinβ€˜πΎ)
cmtbr2.m ∧ = (meetβ€˜πΎ)
cmtbr2.o βŠ₯ = (ocβ€˜πΎ)
cmtbr2.c 𝐢 = (cmβ€˜πΎ)
Assertion
Ref Expression
cmtbr3N ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)))
Proof of Theorem cmtbr3N
StepHypRef Expression
1 cmtbr2.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 cmtbr2.c . . . . 5 𝐢 = (cmβ€˜πΎ)
31, 2cmtcomN 38422 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ π‘ŒπΆπ‘‹))
4 cmtbr2.j . . . . . 6 ∨ = (joinβ€˜πΎ)
5 cmtbr2.m . . . . . 6 ∧ = (meetβ€˜πΎ)
6 cmtbr2.o . . . . . 6 βŠ₯ = (ocβ€˜πΎ)
71, 4, 5, 6, 2cmtbr2N 38426 . . . . 5 ((𝐾 ∈ OML ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (π‘ŒπΆπ‘‹ ↔ π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
873com23 1126 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘ŒπΆπ‘‹ ↔ π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
93, 8bitrd 278 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
10 oveq2 7419 . . . . . 6 (π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹))) β†’ (𝑋 ∧ π‘Œ) = (𝑋 ∧ ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
1110adantl 482 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 ∧ π‘Œ) = (𝑋 ∧ ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
12 omlol 38413 . . . . . . . . 9 (𝐾 ∈ OML β†’ 𝐾 ∈ OL)
13123ad2ant1 1133 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OL)
14 simp2 1137 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
15 omllat 38415 . . . . . . . . . 10 (𝐾 ∈ OML β†’ 𝐾 ∈ Lat)
16153ad2ant1 1133 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
17 simp3 1138 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
181, 4latjcl 18396 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (π‘Œ ∨ 𝑋) ∈ 𝐡)
1916, 17, 14, 18syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∨ 𝑋) ∈ 𝐡)
20 omlop 38414 . . . . . . . . . . 11 (𝐾 ∈ OML β†’ 𝐾 ∈ OP)
21203ad2ant1 1133 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
221, 6opoccl 38367 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
2321, 14, 22syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
241, 4latjcl 18396 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ π‘Œ ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡) β†’ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)) ∈ 𝐡)
2516, 17, 23, 24syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)) ∈ 𝐡)
261, 5latmassOLD 38402 . . . . . . . 8 ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐡 ∧ (π‘Œ ∨ 𝑋) ∈ 𝐡 ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)) ∈ 𝐡)) β†’ ((𝑋 ∧ (π‘Œ ∨ 𝑋)) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹))) = (𝑋 ∧ ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
2713, 14, 19, 25, 26syl13anc 1372 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ (π‘Œ ∨ 𝑋)) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹))) = (𝑋 ∧ ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))))
281, 4latjcom 18404 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (π‘Œ ∨ 𝑋) = (𝑋 ∨ π‘Œ))
2916, 17, 14, 28syl3anc 1371 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∨ 𝑋) = (𝑋 ∨ π‘Œ))
3029oveq2d 7427 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑋)) = (𝑋 ∧ (𝑋 ∨ π‘Œ)))
311, 4, 5latabs2 18433 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ (𝑋 ∨ π‘Œ)) = 𝑋)
3215, 31syl3an1 1163 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ (𝑋 ∨ π‘Œ)) = 𝑋)
3330, 32eqtrd 2772 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑋)) = 𝑋)
341, 4latjcom 18404 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ π‘Œ ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡) β†’ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)) = (( βŠ₯ β€˜π‘‹) ∨ π‘Œ))
3516, 17, 23, 34syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)) = (( βŠ₯ β€˜π‘‹) ∨ π‘Œ))
3633, 35oveq12d 7429 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ (π‘Œ ∨ 𝑋)) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹))) = (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)))
3727, 36eqtr3d 2774 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))) = (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)))
3837adantr 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 ∧ ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))) = (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)))
3911, 38eqtr2d 2773 . . . 4 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ))
4039ex 413 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ = ((π‘Œ ∨ 𝑋) ∧ (π‘Œ ∨ ( βŠ₯ β€˜π‘‹))) β†’ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)))
419, 40sylbid 239 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ β†’ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)))
42 simp1 1136 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OML)
431, 6opoccl 38367 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
4421, 17, 43syl2anc 584 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
451, 5latmcl 18397 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
4616, 14, 44, 45syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
4742, 46, 143jca 1128 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝐾 ∈ OML ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
48 eqid 2732 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
491, 48, 5latmle1 18421 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))(leβ€˜πΎ)𝑋)
5016, 14, 44, 49syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))(leβ€˜πΎ)𝑋)
511, 48, 4, 5, 6omllaw2N 38417 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑋 ∧ ( βŠ₯ β€˜π‘Œ))(leβ€˜πΎ)𝑋 β†’ ((𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋)) = 𝑋))
5247, 50, 51sylc 65 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋)) = 𝑋)
531, 6opoccl 38367 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
5421, 46, 53syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
551, 5latmcl 18397 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∈ 𝐡)
5616, 54, 14, 55syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∈ 𝐡)
571, 4latjcom 18404 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡 ∧ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∈ 𝐡) β†’ ((𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋)) = ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
5816, 46, 56, 57syl3anc 1371 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋)) = ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
5952, 58eqtr3d 2774 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 = ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
6059adantr 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)) β†’ 𝑋 = ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
611, 4, 5, 6oldmm3N 38392 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) = (( βŠ₯ β€˜π‘‹) ∨ π‘Œ))
6212, 61syl3an1 1163 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) = (( βŠ₯ β€˜π‘‹) ∨ π‘Œ))
6362oveq2d 7427 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ)))) = (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)))
641, 5latmcom 18420 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ)))) = (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋))
6516, 14, 54, 64syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ ( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ)))) = (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋))
6663, 65eqtr3d 2774 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋))
6766eqeq1d 2734 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ) ↔ (( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) = (𝑋 ∧ π‘Œ)))
68 oveq1 7418 . . . . . . 7 ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) = (𝑋 ∧ π‘Œ) β†’ ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
6967, 68syl6bi 252 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ) β†’ ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)))))
7069imp 407 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)) β†’ ((( βŠ₯ β€˜(𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) ∧ 𝑋) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
7160, 70eqtrd 2772 . . . 4 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)) β†’ 𝑋 = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ))))
7271ex 413 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ) β†’ 𝑋 = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)))))
731, 4, 5, 6, 2cmtvalN 38384 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋 = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ ( βŠ₯ β€˜π‘Œ)))))
7472, 73sylibrd 258 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ) β†’ π‘‹πΆπ‘Œ))
7541, 74impbid 211 1 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  lecple 17208  occoc 17209  joincjn 18268  meetcmee 18269  Latclat 18388  OPcops 38345  cmccmtN 38346  OLcol 38347  OMLcoml 38348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-lat 18389  df-oposet 38349  df-cmtN 38350  df-ol 38351  df-oml 38352
This theorem is referenced by:  cmtbr4N  38428  omlfh1N  38431
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