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Theorem flfcnp2 23955
Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
flfcnp2.j (𝜑𝐽 ∈ (TopOn‘𝑋))
flfcnp2.k (𝜑𝐾 ∈ (TopOn‘𝑌))
flfcnp2.l (𝜑𝐿 ∈ (Fil‘𝑍))
flfcnp2.a ((𝜑𝑥𝑍) → 𝐴𝑋)
flfcnp2.b ((𝜑𝑥𝑍) → 𝐵𝑌)
flfcnp2.r (𝜑𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥𝑍𝐴)))
flfcnp2.s (𝜑𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥𝑍𝐵)))
flfcnp2.o (𝜑𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))
Assertion
Ref Expression
flfcnp2 (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
Distinct variable groups:   𝑥,𝑂   𝜑,𝑥   𝑥,𝑍   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑅(𝑥)   𝑆(𝑥)   𝐽(𝑥)   𝐾(𝑥)   𝐿(𝑥)   𝑁(𝑥)

Proof of Theorem flfcnp2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7422 . 2 (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2 flfcnp2.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 flfcnp2.k . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 23539 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 582 . . . 4 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6 flfcnp2.l . . . 4 (𝜑𝐿 ∈ (Fil‘𝑍))
7 flfcnp2.a . . . . . 6 ((𝜑𝑥𝑍) → 𝐴𝑋)
8 flfcnp2.b . . . . . 6 ((𝜑𝑥𝑍) → 𝐵𝑌)
97, 8opelxpd 5717 . . . . 5 ((𝜑𝑥𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
109fmpttd 7124 . . . 4 (𝜑 → (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩):𝑍⟶(𝑋 × 𝑌))
11 flfcnp2.r . . . . . 6 (𝜑𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥𝑍𝐴)))
12 flfcnp2.s . . . . . 6 (𝜑𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥𝑍𝐵)))
137fmpttd 7124 . . . . . . 7 (𝜑 → (𝑥𝑍𝐴):𝑍𝑋)
148fmpttd 7124 . . . . . . 7 (𝜑 → (𝑥𝑍𝐵):𝑍𝑌)
15 nfcv 2891 . . . . . . . 8 𝑦⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩
16 nffvmpt1 6907 . . . . . . . . 9 𝑥((𝑥𝑍𝐴)‘𝑦)
17 nffvmpt1 6907 . . . . . . . . 9 𝑥((𝑥𝑍𝐵)‘𝑦)
1816, 17nfop 4891 . . . . . . . 8 𝑥⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩
19 fveq2 6896 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐴)‘𝑥) = ((𝑥𝑍𝐴)‘𝑦))
20 fveq2 6896 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐵)‘𝑥) = ((𝑥𝑍𝐵)‘𝑦))
2119, 20opeq12d 4883 . . . . . . . 8 (𝑥 = 𝑦 → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
2215, 18, 21cbvmpt 5260 . . . . . . 7 (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑦𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
232, 3, 6, 13, 14, 22txflf 23954 . . . . . 6 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥𝑍𝐴)) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥𝑍𝐵)))))
2411, 12, 23mpbir2and 711 . . . . 5 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)))
25 simpr 483 . . . . . . . . 9 ((𝜑𝑥𝑍) → 𝑥𝑍)
26 eqid 2725 . . . . . . . . . 10 (𝑥𝑍𝐴) = (𝑥𝑍𝐴)
2726fvmpt2 7015 . . . . . . . . 9 ((𝑥𝑍𝐴𝑋) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
2825, 7, 27syl2anc 582 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
29 eqid 2725 . . . . . . . . . 10 (𝑥𝑍𝐵) = (𝑥𝑍𝐵)
3029fvmpt2 7015 . . . . . . . . 9 ((𝑥𝑍𝐵𝑌) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3125, 8, 30syl2anc 582 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3228, 31opeq12d 4883 . . . . . . 7 ((𝜑𝑥𝑍) → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3332mpteq2dva 5249 . . . . . 6 (𝜑 → (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
3433fveq2d 6900 . . . . 5 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)) = (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)))
3524, 34eleqtrd 2827 . . . 4 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)))
36 flfcnp2.o . . . 4 (𝜑𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))
37 flfcnp 23952 . . . 4 ((((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩):𝑍⟶(𝑋 × 𝑌)) ∧ (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)) ∧ 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))) → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))))
385, 6, 10, 35, 36, 37syl32anc 1375 . . 3 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))))
39 eqidd 2726 . . . . 5 (𝜑 → (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
40 cnptop2 23191 . . . . . . . . 9 (𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
4136, 40syl 17 . . . . . . . 8 (𝜑𝑁 ∈ Top)
42 toptopon2 22864 . . . . . . . 8 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
4341, 42sylib 217 . . . . . . 7 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
44 cnpf2 23198 . . . . . . 7 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩)) → 𝑂:(𝑋 × 𝑌)⟶ 𝑁)
455, 43, 36, 44syl3anc 1368 . . . . . 6 (𝜑𝑂:(𝑋 × 𝑌)⟶ 𝑁)
4645feqmptd 6966 . . . . 5 (𝜑𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂𝑦)))
47 fveq2 6896 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
48 df-ov 7422 . . . . . 6 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
4947, 48eqtr4di 2783 . . . . 5 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝐴𝑂𝐵))
509, 39, 46, 49fmptco 7138 . . . 4 (𝜑 → (𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥𝑍 ↦ (𝐴𝑂𝐵)))
5150fveq2d 6900 . . 3 (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
5238, 51eleqtrd 2827 . 2 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
531, 52eqeltrid 2829 1 (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4636   cuni 4909  cmpt 5232   × cxp 5676  ccom 5682  wf 6545  cfv 6549  (class class class)co 7419  Topctop 22839  TopOnctopon 22856   CnP ccnp 23173   ×t ctx 23508  Filcfil 23793   fLimf cflf 23883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  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 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-topgen 17428  df-fbas 21293  df-fg 21294  df-top 22840  df-topon 22857  df-bases 22893  df-ntr 22968  df-nei 23046  df-cnp 23176  df-tx 23510  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888
This theorem is referenced by:  tsmsadd  24095
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