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Theorem flfcnp2 22607
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 7151 . 2 (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2 flfcnp2.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 flfcnp2.k . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 22191 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 586 . . . 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 5586 . . . . 5 ((𝜑𝑥𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
109fmpttd 6872 . . . 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 6872 . . . . . . 7 (𝜑 → (𝑥𝑍𝐴):𝑍𝑋)
148fmpttd 6872 . . . . . . 7 (𝜑 → (𝑥𝑍𝐵):𝑍𝑌)
15 nfcv 2975 . . . . . . . 8 𝑦⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩
16 nffvmpt1 6674 . . . . . . . . 9 𝑥((𝑥𝑍𝐴)‘𝑦)
17 nffvmpt1 6674 . . . . . . . . 9 𝑥((𝑥𝑍𝐵)‘𝑦)
1816, 17nfop 4811 . . . . . . . 8 𝑥⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩
19 fveq2 6663 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐴)‘𝑥) = ((𝑥𝑍𝐴)‘𝑦))
20 fveq2 6663 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐵)‘𝑥) = ((𝑥𝑍𝐵)‘𝑦))
2119, 20opeq12d 4803 . . . . . . . 8 (𝑥 = 𝑦 → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
2215, 18, 21cbvmpt 5158 . . . . . . 7 (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑦𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
232, 3, 6, 13, 14, 22txflf 22606 . . . . . 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 487 . . . . . . . . 9 ((𝜑𝑥𝑍) → 𝑥𝑍)
26 eqid 2819 . . . . . . . . . 10 (𝑥𝑍𝐴) = (𝑥𝑍𝐴)
2726fvmpt2 6772 . . . . . . . . 9 ((𝑥𝑍𝐴𝑋) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
2825, 7, 27syl2anc 586 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
29 eqid 2819 . . . . . . . . . 10 (𝑥𝑍𝐵) = (𝑥𝑍𝐵)
3029fvmpt2 6772 . . . . . . . . 9 ((𝑥𝑍𝐵𝑌) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3125, 8, 30syl2anc 586 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3228, 31opeq12d 4803 . . . . . . 7 ((𝜑𝑥𝑍) → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3332mpteq2dva 5152 . . . . . 6 (𝜑 → (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
3433fveq2d 6667 . . . . 5 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)) = (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)))
3524, 34eleqtrd 2913 . . . 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 22604 . . . 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 1372 . . 3 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))))
39 eqidd 2820 . . . . 5 (𝜑 → (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
40 cnptop2 21843 . . . . . . . . 9 (𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
4136, 40syl 17 . . . . . . . 8 (𝜑𝑁 ∈ Top)
42 toptopon2 21518 . . . . . . . 8 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
4341, 42sylib 220 . . . . . . 7 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
44 cnpf2 21850 . . . . . . 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 1365 . . . . . 6 (𝜑𝑂:(𝑋 × 𝑌)⟶ 𝑁)
4645feqmptd 6726 . . . . 5 (𝜑𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂𝑦)))
47 fveq2 6663 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
48 df-ov 7151 . . . . . 6 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
4947, 48syl6eqr 2872 . . . . 5 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝐴𝑂𝐵))
509, 39, 46, 49fmptco 6884 . . . 4 (𝜑 → (𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥𝑍 ↦ (𝐴𝑂𝐵)))
5150fveq2d 6667 . . 3 (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
5238, 51eleqtrd 2913 . 2 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
531, 52eqeltrid 2915 1 (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  cop 4565   cuni 4830  cmpt 5137   × cxp 5546  ccom 5552  wf 6344  cfv 6348  (class class class)co 7148  Topctop 21493  TopOnctopon 21510   CnP ccnp 21825   ×t ctx 22160  Filcfil 22445   fLimf cflf 22535
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-topgen 16709  df-fbas 20534  df-fg 20535  df-top 21494  df-topon 21511  df-bases 21546  df-ntr 21620  df-nei 21698  df-cnp 21828  df-tx 22162  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540
This theorem is referenced by:  tsmsadd  22747
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