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Theorem flfcnp2 22012
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 6816 . 2 (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2 flfcnp2.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 flfcnp2.k . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 21596 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
52, 3, 4syl2anc 696 . . . 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)
9 opelxpi 5305 . . . . . 6 ((𝐴𝑋𝐵𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
107, 8, 9syl2anc 696 . . . . 5 ((𝜑𝑥𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11 eqid 2760 . . . . 5 (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)
1210, 11fmptd 6548 . . . 4 (𝜑 → (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩):𝑍⟶(𝑋 × 𝑌))
13 flfcnp2.r . . . . . 6 (𝜑𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥𝑍𝐴)))
14 flfcnp2.s . . . . . 6 (𝜑𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥𝑍𝐵)))
15 eqid 2760 . . . . . . . 8 (𝑥𝑍𝐴) = (𝑥𝑍𝐴)
167, 15fmptd 6548 . . . . . . 7 (𝜑 → (𝑥𝑍𝐴):𝑍𝑋)
17 eqid 2760 . . . . . . . 8 (𝑥𝑍𝐵) = (𝑥𝑍𝐵)
188, 17fmptd 6548 . . . . . . 7 (𝜑 → (𝑥𝑍𝐵):𝑍𝑌)
19 nfcv 2902 . . . . . . . 8 𝑦⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩
20 nffvmpt1 6360 . . . . . . . . 9 𝑥((𝑥𝑍𝐴)‘𝑦)
21 nffvmpt1 6360 . . . . . . . . 9 𝑥((𝑥𝑍𝐵)‘𝑦)
2220, 21nfop 4569 . . . . . . . 8 𝑥⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩
23 fveq2 6352 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐴)‘𝑥) = ((𝑥𝑍𝐴)‘𝑦))
24 fveq2 6352 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐵)‘𝑥) = ((𝑥𝑍𝐵)‘𝑦))
2523, 24opeq12d 4561 . . . . . . . 8 (𝑥 = 𝑦 → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
2619, 22, 25cbvmpt 4901 . . . . . . 7 (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑦𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
272, 3, 6, 16, 18, 26txflf 22011 . . . . . 6 (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥𝑍𝐴)) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥𝑍𝐵)))))
2813, 14, 27mpbir2and 995 . . . . 5 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)))
29 simpr 479 . . . . . . . . 9 ((𝜑𝑥𝑍) → 𝑥𝑍)
3015fvmpt2 6453 . . . . . . . . 9 ((𝑥𝑍𝐴𝑋) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
3129, 7, 30syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
3217fvmpt2 6453 . . . . . . . . 9 ((𝑥𝑍𝐵𝑌) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3329, 8, 32syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3431, 33opeq12d 4561 . . . . . . 7 ((𝜑𝑥𝑍) → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3534mpteq2dva 4896 . . . . . 6 (𝜑 → (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
3635fveq2d 6356 . . . . 5 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)) = (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)))
3728, 36eleqtrd 2841 . . . 4 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)))
38 flfcnp2.o . . . 4 (𝜑𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))
39 flfcnp 22009 . . . 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⟩))))
405, 6, 12, 37, 38, 39syl32anc 1485 . . 3 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))))
41 eqidd 2761 . . . . 5 (𝜑 → (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
42 cnptop2 21249 . . . . . . . . 9 (𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
4338, 42syl 17 . . . . . . . 8 (𝜑𝑁 ∈ Top)
44 eqid 2760 . . . . . . . . 9 𝑁 = 𝑁
4544toptopon 20924 . . . . . . . 8 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
4643, 45sylib 208 . . . . . . 7 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
47 cnpf2 21256 . . . . . . 7 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑁 ∈ (TopOn‘ 𝑁) ∧ 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩)) → 𝑂:(𝑋 × 𝑌)⟶ 𝑁)
485, 46, 38, 47syl3anc 1477 . . . . . 6 (𝜑𝑂:(𝑋 × 𝑌)⟶ 𝑁)
4948feqmptd 6411 . . . . 5 (𝜑𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂𝑦)))
50 fveq2 6352 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
51 df-ov 6816 . . . . . 6 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
5250, 51syl6eqr 2812 . . . . 5 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝐴𝑂𝐵))
5310, 41, 49, 52fmptco 6559 . . . 4 (𝜑 → (𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥𝑍 ↦ (𝐴𝑂𝐵)))
5453fveq2d 6356 . . 3 (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
5540, 54eleqtrd 2841 . 2 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
561, 55syl5eqel 2843 1 (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cop 4327   cuni 4588  cmpt 4881   × cxp 5264  ccom 5270  wf 6045  cfv 6049  (class class class)co 6813  Topctop 20900  TopOnctopon 20917   CnP ccnp 21231   ×t ctx 21565  Filcfil 21850   fLimf cflf 21940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-topgen 16306  df-fbas 19945  df-fg 19946  df-top 20901  df-topon 20918  df-bases 20952  df-ntr 21026  df-nei 21104  df-cnp 21234  df-tx 21567  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945
This theorem is referenced by:  tsmsadd  22151
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