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Theorem linepsubN 39735
Description: A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
linepsub.n 𝑁 = (Lines‘𝐾)
linepsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
linepsubN ((𝐾 ∈ Lat ∧ 𝑋𝑁) → 𝑋𝑆)

Proof of Theorem linepsubN
Dummy variables 𝑎 𝑏 𝑐 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 4090 . . . . . . . 8 {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)
2 sseq1 4021 . . . . . . . 8 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑋 ⊆ (Atoms‘𝐾) ↔ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)))
31, 2mpbiri 258 . . . . . . 7 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → 𝑋 ⊆ (Atoms‘𝐾))
43a1i 11 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → 𝑋 ⊆ (Atoms‘𝐾)))
5 eqid 2735 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2735 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
75, 6atbase 39271 . . . . . . . . 9 (𝑎 ∈ (Atoms‘𝐾) → 𝑎 ∈ (Base‘𝐾))
85, 6atbase 39271 . . . . . . . . 9 (𝑏 ∈ (Atoms‘𝐾) → 𝑏 ∈ (Base‘𝐾))
97, 8anim12i 613 . . . . . . . 8 ((𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾)) → (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)))
10 eqid 2735 . . . . . . . . . 10 (join‘𝐾) = (join‘𝐾)
115, 10latjcl 18497 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
12113expb 1119 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
139, 12sylan2 593 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
14 eleq2 2828 . . . . . . . . . . . . . . . . . . 19 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑝𝑋𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}))
15 breq1 5151 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑝 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
1615elrab 3695 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
175, 6atbase 39271 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1817anim1i 615 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
1916, 18sylbi 217 . . . . . . . . . . . . . . . . . . 19 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
2014, 19biimtrdi 253 . . . . . . . . . . . . . . . . . 18 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑝𝑋 → (𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
21 eleq2 2828 . . . . . . . . . . . . . . . . . . 19 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑞𝑋𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}))
22 breq1 5151 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑞 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
2322elrab 3695 . . . . . . . . . . . . . . . . . . . 20 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
245, 6atbase 39271 . . . . . . . . . . . . . . . . . . . . 21 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
2524anim1i 615 . . . . . . . . . . . . . . . . . . . 20 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
2623, 25sylbi 217 . . . . . . . . . . . . . . . . . . 19 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
2721, 26biimtrdi 253 . . . . . . . . . . . . . . . . . 18 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑞𝑋 → (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
2820, 27anim12d 609 . . . . . . . . . . . . . . . . 17 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ((𝑝𝑋𝑞𝑋) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
29 an4 656 . . . . . . . . . . . . . . . . 17 (((𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))) ↔ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
3028, 29imbitrdi 251 . . . . . . . . . . . . . . . 16 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ((𝑝𝑋𝑞𝑋) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
3130imp 406 . . . . . . . . . . . . . . 15 ((𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ∧ (𝑝𝑋𝑞𝑋)) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
3231anim2i 617 . . . . . . . . . . . . . 14 (((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ∧ (𝑝𝑋𝑞𝑋))) → ((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
3332anassrs 467 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) → ((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
345, 6atbase 39271 . . . . . . . . . . . . 13 (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
35 eqid 2735 . . . . . . . . . . . . . . . . . . . . 21 (le‘𝐾) = (le‘𝐾)
365, 35, 10latjle12 18508 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ Lat ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾) ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
3736biimpd 229 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ Lat ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾) ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
38373exp2 1353 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ Lat → (𝑝 ∈ (Base‘𝐾) → (𝑞 ∈ (Base‘𝐾) → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
3938impd 410 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ Lat → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
4039com23 86 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Lat → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
4140imp43 427 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏))
4241adantr 480 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏))
435, 10latjcl 18497 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
44433expib 1121 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ Lat → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)))
455, 35lattr 18502 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ Lat ∧ (𝑟 ∈ (Base‘𝐾) ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
46453exp2 1353 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ Lat → (𝑟 ∈ (Base‘𝐾) → ((𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
4746com24 95 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ Lat → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) → (𝑟 ∈ (Base‘𝐾) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
4844, 47syl5d 73 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Lat → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑟 ∈ (Base‘𝐾) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
4948imp41 425 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾))) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
5049adantlrr 721 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
5142, 50mpan2d 694 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
5233, 34, 51syl2an 596 . . . . . . . . . . . 12 (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
53 simpr 484 . . . . . . . . . . . 12 (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Atoms‘𝐾))
5452, 53jctild 525 . . . . . . . . . . 11 (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
55 eleq2 2828 . . . . . . . . . . . . 13 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑟𝑋𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}))
56 breq1 5151 . . . . . . . . . . . . . 14 (𝑐 = 𝑟 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
5756elrab 3695 . . . . . . . . . . . . 13 (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
5855, 57bitrdi 287 . . . . . . . . . . . 12 (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑟𝑋 ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
5958ad3antlr 731 . . . . . . . . . . 11 (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟𝑋 ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
6054, 59sylibrd 259 . . . . . . . . . 10 (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))
6160ralrimiva 3144 . . . . . . . . 9 ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝𝑋𝑞𝑋)) → ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))
6261ralrimivva 3200 . . . . . . . 8 (((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) → ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))
6362ex 412 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋)))
6413, 63syldan 591 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋)))
654, 64jcad 512 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
6665adantld 490 . . . 4 ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → ((𝑎𝑏𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
6766rexlimdvva 3211 . . 3 (𝐾 ∈ Lat → (∃𝑎 ∈ (Atoms‘𝐾)∃𝑏 ∈ (Atoms‘𝐾)(𝑎𝑏𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
68 linepsub.n . . . 4 𝑁 = (Lines‘𝐾)
6935, 10, 6, 68isline 39722 . . 3 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑎 ∈ (Atoms‘𝐾)∃𝑏 ∈ (Atoms‘𝐾)(𝑎𝑏𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)})))
70 linepsub.s . . . 4 𝑆 = (PSubSp‘𝐾)
7135, 10, 6, 70ispsubsp 39728 . . 3 (𝐾 ∈ Lat → (𝑋𝑆 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝𝑋𝑞𝑋𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
7267, 69, 713imtr4d 294 . 2 (𝐾 ∈ Lat → (𝑋𝑁𝑋𝑆))
7372imp 406 1 ((𝐾 ∈ Lat ∧ 𝑋𝑁) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  Linesclines 39477  PSubSpcpsubsp 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-ats 39249  df-lines 39484  df-psubsp 39486
This theorem is referenced by: (None)
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