Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limccog Structured version   Visualization version   GIF version

Theorem limccog 43868
Description: Limit of the composition of two functions. If the limit of 𝐹 at 𝐴 is 𝐡 and the limit of 𝐺 at 𝐡 is 𝐢, then the limit of 𝐺 ∘ 𝐹 at 𝐴 is 𝐢. With respect to limcco 25260 and limccnp 25258, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
limccog.1 (πœ‘ β†’ ran 𝐹 βŠ† (dom 𝐺 βˆ– {𝐡}))
limccog.2 (πœ‘ β†’ 𝐡 ∈ (𝐹 limβ„‚ 𝐴))
limccog.3 (πœ‘ β†’ 𝐢 ∈ (𝐺 limβ„‚ 𝐡))
Assertion
Ref Expression
limccog (πœ‘ β†’ 𝐢 ∈ ((𝐺 ∘ 𝐹) limβ„‚ 𝐴))

Proof of Theorem limccog
Dummy variables 𝑒 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccl 25242 . . 3 (𝐺 limβ„‚ 𝐡) βŠ† β„‚
2 limccog.3 . . 3 (πœ‘ β†’ 𝐢 ∈ (𝐺 limβ„‚ 𝐡))
31, 2sselid 3943 . 2 (πœ‘ β†’ 𝐢 ∈ β„‚)
4 limcrcl 25241 . . . . . . . . . . . 12 (𝐢 ∈ (𝐺 limβ„‚ 𝐡) β†’ (𝐺:dom πΊβŸΆβ„‚ ∧ dom 𝐺 βŠ† β„‚ ∧ 𝐡 ∈ β„‚))
52, 4syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (𝐺:dom πΊβŸΆβ„‚ ∧ dom 𝐺 βŠ† β„‚ ∧ 𝐡 ∈ β„‚))
65simp1d 1143 . . . . . . . . . 10 (πœ‘ β†’ 𝐺:dom πΊβŸΆβ„‚)
75simp2d 1144 . . . . . . . . . 10 (πœ‘ β†’ dom 𝐺 βŠ† β„‚)
85simp3d 1145 . . . . . . . . . 10 (πœ‘ β†’ 𝐡 ∈ β„‚)
9 eqid 2737 . . . . . . . . . 10 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
106, 7, 8, 9ellimc2 25244 . . . . . . . . 9 (πœ‘ β†’ (𝐢 ∈ (𝐺 limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)))))
112, 10mpbid 231 . . . . . . . 8 (πœ‘ β†’ (𝐢 ∈ β„‚ ∧ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒))))
1211simprd 497 . . . . . . 7 (πœ‘ β†’ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)))
1312r19.21bi 3235 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) β†’ (𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)))
1413imp 408 . . . . 5 (((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒))
15 simp1ll 1237 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ πœ‘)
16 simp2 1138 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ 𝑣 ∈ (TopOpenβ€˜β„‚fld))
17 simp3l 1202 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ 𝐡 ∈ 𝑣)
18 limccog.2 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐡 ∈ (𝐹 limβ„‚ 𝐴))
19 limcrcl 25241 . . . . . . . . . . . . . . 15 (𝐡 ∈ (𝐹 limβ„‚ 𝐴) β†’ (𝐹:dom πΉβŸΆβ„‚ ∧ dom 𝐹 βŠ† β„‚ ∧ 𝐴 ∈ β„‚))
2018, 19syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝐹:dom πΉβŸΆβ„‚ ∧ dom 𝐹 βŠ† β„‚ ∧ 𝐴 ∈ β„‚))
2120simp1d 1143 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„‚)
2220simp2d 1144 . . . . . . . . . . . . 13 (πœ‘ β†’ dom 𝐹 βŠ† β„‚)
2320simp3d 1145 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐴 ∈ β„‚)
2421, 22, 23, 9ellimc2 25244 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐡 ∈ (𝐹 limβ„‚ 𝐴) ↔ (𝐡 ∈ β„‚ ∧ βˆ€π‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣)))))
2518, 24mpbid 231 . . . . . . . . . . 11 (πœ‘ β†’ (𝐡 ∈ β„‚ ∧ βˆ€π‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣))))
2625simprd 497 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣)))
2726r19.21bi 3235 . . . . . . . . 9 ((πœ‘ ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld)) β†’ (𝐡 ∈ 𝑣 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣)))
2827imp 408 . . . . . . . 8 (((πœ‘ ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐡 ∈ 𝑣) β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣))
2915, 16, 17, 28syl21anc 837 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣))
30 imaco 6204 . . . . . . . . . . 11 ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) = (𝐺 β€œ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))))
3115ad2antrr 725 . . . . . . . . . . . 12 ((((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ πœ‘)
32 simpl3r 1230 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) β†’ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)
3332adantr 482 . . . . . . . . . . . 12 ((((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)
34 simpr 486 . . . . . . . . . . . 12 ((((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣)
35 simpr 486 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣)
36 imassrn 6025 . . . . . . . . . . . . . . . . . 18 (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† ran 𝐹
37 limccog.1 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ ran 𝐹 βŠ† (dom 𝐺 βˆ– {𝐡}))
3836, 37sstrid 3956 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† (dom 𝐺 βˆ– {𝐡}))
3938adantr 482 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† (dom 𝐺 βˆ– {𝐡}))
4035, 39ssind 4193 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† (𝑣 ∩ (dom 𝐺 βˆ– {𝐡})))
41 imass2 6055 . . . . . . . . . . . . . . 15 ((𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† (𝑣 ∩ (dom 𝐺 βˆ– {𝐡})) β†’ (𝐺 β€œ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴})))) βŠ† (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))))
4240, 41syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐺 β€œ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴})))) βŠ† (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))))
4342adantlr 714 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐺 β€œ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴})))) βŠ† (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))))
44 simplr 768 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)
4543, 44sstrd 3955 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐺 β€œ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴})))) βŠ† 𝑒)
4631, 33, 34, 45syl21anc 837 . . . . . . . . . . 11 ((((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐺 β€œ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴})))) βŠ† 𝑒)
4730, 46eqsstrid 3993 . . . . . . . . . 10 ((((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)
4847ex 414 . . . . . . . . 9 (((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) β†’ ((𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣 β†’ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒))
4948anim2d 613 . . . . . . . 8 (((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) ∧ 𝑀 ∈ (TopOpenβ€˜β„‚fld)) β†’ ((𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ (𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)))
5049reximdva 3166 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ (βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ (𝐹 β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑣) β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)))
5129, 50mpd 15 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒))
5251rexlimdv3a 3157 . . . . 5 (((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) β†’ (βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐺 β€œ (𝑣 ∩ (dom 𝐺 βˆ– {𝐡}))) βŠ† 𝑒) β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)))
5314, 52mpd 15 . . . 4 (((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) ∧ 𝐢 ∈ 𝑒) β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒))
5453ex 414 . . 3 ((πœ‘ ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) β†’ (𝐢 ∈ 𝑒 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)))
5554ralrimiva 3144 . 2 (πœ‘ β†’ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)))
5621ffund 6673 . . . . . 6 (πœ‘ β†’ Fun 𝐹)
57 fdmrn 6701 . . . . . 6 (Fun 𝐹 ↔ 𝐹:dom 𝐹⟢ran 𝐹)
5856, 57sylib 217 . . . . 5 (πœ‘ β†’ 𝐹:dom 𝐹⟢ran 𝐹)
5937difss2d 4095 . . . . 5 (πœ‘ β†’ ran 𝐹 βŠ† dom 𝐺)
6058, 59fssd 6687 . . . 4 (πœ‘ β†’ 𝐹:dom 𝐹⟢dom 𝐺)
61 fco 6693 . . . 4 ((𝐺:dom πΊβŸΆβ„‚ ∧ 𝐹:dom 𝐹⟢dom 𝐺) β†’ (𝐺 ∘ 𝐹):dom πΉβŸΆβ„‚)
626, 60, 61syl2anc 585 . . 3 (πœ‘ β†’ (𝐺 ∘ 𝐹):dom πΉβŸΆβ„‚)
6362, 22, 23, 9ellimc2 25244 . 2 (πœ‘ β†’ (𝐢 ∈ ((𝐺 ∘ 𝐹) limβ„‚ 𝐴) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘€ ∈ (TopOpenβ€˜β„‚fld)(𝐴 ∈ 𝑀 ∧ ((𝐺 ∘ 𝐹) β€œ (𝑀 ∩ (dom 𝐹 βˆ– {𝐴}))) βŠ† 𝑒)))))
643, 55, 63mpbir2and 712 1 (πœ‘ β†’ 𝐢 ∈ ((𝐺 ∘ 𝐹) limβ„‚ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074   βˆ– cdif 3908   ∩ cin 3910   βŠ† wss 3911  {csn 4587  dom cdm 5634  ran crn 5635   β€œ cima 5637   ∘ ccom 5638  Fun wfun 6491  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11050  TopOpenctopn 17304  β„‚fldccnfld 20799   limβ„‚ climc 25229
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-map 8768  df-pm 8769  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fi 9348  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-7 12222  df-8 12223  df-9 12224  df-n0 12415  df-z 12501  df-dec 12620  df-uz 12765  df-q 12875  df-rp 12917  df-xneg 13034  df-xadd 13035  df-xmul 13036  df-fz 13426  df-seq 13908  df-exp 13969  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-struct 17020  df-slot 17055  df-ndx 17067  df-base 17085  df-plusg 17147  df-mulr 17148  df-starv 17149  df-tset 17153  df-ple 17154  df-ds 17156  df-unif 17157  df-rest 17305  df-topn 17306  df-topgen 17326  df-psmet 20791  df-xmet 20792  df-met 20793  df-bl 20794  df-mopn 20795  df-cnfld 20800  df-top 22246  df-topon 22263  df-topsp 22285  df-bases 22299  df-cnp 22582  df-xms 23676  df-ms 23677  df-limc 25233
This theorem is referenced by:  dirkercncflem2  44352  fourierdlem53  44407  fourierdlem93  44447  fourierdlem111  44465
  Copyright terms: Public domain W3C validator