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Theorem flfcnp2 22033
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 6818 . 2 (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2 flfcnp2.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 flfcnp2.k . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 txtopon 21617 . . . . 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 5306 . . . . . 6 ((𝐴𝑋𝐵𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
107, 8, 9syl2anc 696 . . . . 5 ((𝜑𝑥𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11 eqid 2761 . . . . 5 (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)
1210, 11fmptd 6550 . . . 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 2761 . . . . . . . 8 (𝑥𝑍𝐴) = (𝑥𝑍𝐴)
167, 15fmptd 6550 . . . . . . 7 (𝜑 → (𝑥𝑍𝐴):𝑍𝑋)
17 eqid 2761 . . . . . . . 8 (𝑥𝑍𝐵) = (𝑥𝑍𝐵)
188, 17fmptd 6550 . . . . . . 7 (𝜑 → (𝑥𝑍𝐵):𝑍𝑌)
19 nfcv 2903 . . . . . . . 8 𝑦⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩
20 nffvmpt1 6362 . . . . . . . . 9 𝑥((𝑥𝑍𝐴)‘𝑦)
21 nffvmpt1 6362 . . . . . . . . 9 𝑥((𝑥𝑍𝐵)‘𝑦)
2220, 21nfop 4570 . . . . . . . 8 𝑥⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩
23 fveq2 6354 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐴)‘𝑥) = ((𝑥𝑍𝐴)‘𝑦))
24 fveq2 6354 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥𝑍𝐵)‘𝑥) = ((𝑥𝑍𝐵)‘𝑦))
2523, 24opeq12d 4562 . . . . . . . 8 (𝑥 = 𝑦 → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
2619, 22, 25cbvmpt 4902 . . . . . . 7 (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑦𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑦), ((𝑥𝑍𝐵)‘𝑦)⟩)
272, 3, 6, 16, 18, 26txflf 22032 . . . . . 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 6455 . . . . . . . . 9 ((𝑥𝑍𝐴𝑋) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
3129, 7, 30syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐴)‘𝑥) = 𝐴)
3217fvmpt2 6455 . . . . . . . . 9 ((𝑥𝑍𝐵𝑌) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3329, 8, 32syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝑍) → ((𝑥𝑍𝐵)‘𝑥) = 𝐵)
3431, 33opeq12d 4562 . . . . . . 7 ((𝜑𝑥𝑍) → ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3534mpteq2dva 4897 . . . . . 6 (𝜑 → (𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
3635fveq2d 6358 . . . . 5 (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨((𝑥𝑍𝐴)‘𝑥), ((𝑥𝑍𝐵)‘𝑥)⟩)) = (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)))
3728, 36eleqtrd 2842 . . . 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 22030 . . . 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 2762 . . . . 5 (𝜑 → (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))
42 cnptop2 21270 . . . . . . . . 9 (𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
4338, 42syl 17 . . . . . . . 8 (𝜑𝑁 ∈ Top)
44 eqid 2761 . . . . . . . . 9 𝑁 = 𝑁
4544toptopon 20945 . . . . . . . 8 (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘ 𝑁))
4643, 45sylib 208 . . . . . . 7 (𝜑𝑁 ∈ (TopOn‘ 𝑁))
47 cnpf2 21277 . . . . . . 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 6413 . . . . 5 (𝜑𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂𝑦)))
50 fveq2 6354 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
51 df-ov 6818 . . . . . 6 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
5250, 51syl6eqr 2813 . . . . 5 (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂𝑦) = (𝐴𝑂𝐵))
5310, 41, 49, 52fmptco 6561 . . . 4 (𝜑 → (𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥𝑍 ↦ (𝐴𝑂𝐵)))
5453fveq2d 6358 . . 3 (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
5540, 54eleqtrd 2842 . 2 (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
561, 55syl5eqel 2844 1 (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  cop 4328   cuni 4589  cmpt 4882   × cxp 5265  ccom 5271  wf 6046  cfv 6050  (class class class)co 6815  Topctop 20921  TopOnctopon 20938   CnP ccnp 21252   ×t ctx 21586  Filcfil 21871   fLimf cflf 21961
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-map 8028  df-topgen 16327  df-fbas 19966  df-fg 19967  df-top 20922  df-topon 20939  df-bases 20973  df-ntr 21047  df-nei 21125  df-cnp 21255  df-tx 21588  df-fil 21872  df-fm 21964  df-flim 21965  df-flf 21966
This theorem is referenced by:  tsmsadd  22172
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