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Theorem unconn 23437
Description: The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
Assertion
Ref Expression
unconn ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ≠ ∅) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))

Proof of Theorem unconn
Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4353 . . 3 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 uniiun 5058 . . . . . . . . 9 {𝐴, 𝐵} = 𝑘 ∈ {𝐴, 𝐵}𝑘
3 simpl1 1192 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 22932 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
53, 4syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑋𝐽)
6 simpl2l 1227 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐴𝑋)
75, 6ssexd 5324 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐴 ∈ V)
8 simpl2r 1228 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵𝑋)
95, 8ssexd 5324 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10 uniprg 4923 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
117, 9, 10syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → {𝐴, 𝐵} = (𝐴𝐵))
122, 11eqtr3id 2791 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑘 ∈ {𝐴, 𝐵}𝑘 = (𝐴𝐵))
1312oveq2d 7447 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) = (𝐽t (𝐴𝐵)))
14 vex 3484 . . . . . . . . . 10 𝑘 ∈ V
1514elpr 4650 . . . . . . . . 9 (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴𝑘 = 𝐵))
16 simpl2 1193 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐴𝑋𝐵𝑋))
17 sseq1 4009 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑘𝑋𝐴𝑋))
1817biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝐴𝑋𝑘𝑋))
19 sseq1 4009 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑘𝑋𝐵𝑋))
2019biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝐵𝑋𝑘𝑋))
2118, 20jaoa 958 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝐴𝑋𝐵𝑋) → 𝑘𝑋))
2216, 21mpan9 506 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑘𝑋)
2315, 22sylan2b 594 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑘𝑋)
24 simpl3 1194 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑥 ∈ (𝐴𝐵))
25 elin 3967 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2624, 25sylib 218 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝑥𝐴𝑥𝐵))
27 eleq2 2830 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑥𝑘𝑥𝐴))
2827biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝑥𝐴𝑥𝑘))
29 eleq2 2830 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑥𝑘𝑥𝐵))
3029biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝑥𝐵𝑥𝑘))
3128, 30jaoa 958 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝑥𝐴𝑥𝐵) → 𝑥𝑘))
3226, 31mpan9 506 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑥𝑘)
3315, 32sylan2b 594 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑥𝑘)
34 simpr 484 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn))
35 oveq2 7439 . . . . . . . . . . . . 13 (𝑘 = 𝐴 → (𝐽t 𝑘) = (𝐽t 𝐴))
3635eleq1d 2826 . . . . . . . . . . . 12 (𝑘 = 𝐴 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐴) ∈ Conn))
3736biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐴 → ((𝐽t 𝐴) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
38 oveq2 7439 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐽t 𝑘) = (𝐽t 𝐵))
3938eleq1d 2826 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐵) ∈ Conn))
4039biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐽t 𝐵) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
4137, 40jaoa 958 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t 𝑘) ∈ Conn))
4234, 41mpan9 506 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → (𝐽t 𝑘) ∈ Conn)
4315, 42sylan2b 594 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → (𝐽t 𝑘) ∈ Conn)
443, 23, 33, 43iunconn 23436 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) ∈ Conn)
4513, 44eqeltrrd 2842 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t (𝐴𝐵)) ∈ Conn)
4645ex 412 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))
47463expia 1122 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
4847exlimdv 1933 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (∃𝑥 𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
491, 48biimtrid 242 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐵) ≠ ∅ → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
50493impia 1118 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ≠ ∅) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  {cpr 4628   cuni 4907   ciun 4991  cfv 6561  (class class class)co 7431  t crest 17465  TopOnctopon 22916  Conncconn 23419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-conn 23420
This theorem is referenced by: (None)
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