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Mirrors > Home > MPE Home > Th. List > lmcn2 | Structured version Visualization version GIF version |
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, 15-May-2014.) |
Ref | Expression |
---|---|
txlm.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
txlm.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
txlm.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
txlm.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
txlm.f | ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) |
txlm.g | ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) |
lmcn2.fl | ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑅) |
lmcn2.gl | ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐾)𝑆) |
lmcn2.o | ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
lmcn2.h | ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) |
Ref | Expression |
---|---|
lmcn2 | ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txlm.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) | |
2 | 1 | ffvelrnda 6267 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋) |
3 | txlm.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) | |
4 | 3 | ffvelrnda 6267 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌) |
5 | opelxpi 5072 | . . . . . 6 ⊢ (((𝐹‘𝑛) ∈ 𝑋 ∧ (𝐺‘𝑛) ∈ 𝑌) → 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 ∈ (𝑋 × 𝑌)) | |
6 | 2, 4, 5 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 ∈ (𝑋 × 𝑌)) |
7 | eqidd 2611 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
8 | txlm.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
9 | txlm.k | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
10 | txtopon 21204 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
11 | 8, 9, 10 | syl2anc 691 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
12 | lmcn2.o | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | |
13 | cntop2 20855 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁) → 𝑁 ∈ Top) | |
14 | 12, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top) |
15 | eqid 2610 | . . . . . . . . 9 ⊢ ∪ 𝑁 = ∪ 𝑁 | |
16 | 15 | toptopon 20548 | . . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) |
17 | 14, 16 | sylib 207 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
18 | cnf2 20863 | . . . . . . 7 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) | |
19 | 11, 17, 12, 18 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) |
20 | 19 | feqmptd 6159 | . . . . 5 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑥))) |
21 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
22 | df-ov 6552 | . . . . . 6 ⊢ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
23 | 21, 22 | syl6eqr 2662 | . . . . 5 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) |
24 | 6, 7, 20, 23 | fmptco 6303 | . . . 4 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)))) |
25 | lmcn2.h | . . . 4 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) | |
26 | 24, 25 | syl6eqr 2662 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) = 𝐻) |
27 | lmcn2.fl | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑅) | |
28 | lmcn2.gl | . . . . 5 ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐾)𝑆) | |
29 | txlm.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
30 | txlm.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
31 | eqid 2610 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
32 | 29, 30, 8, 9, 1, 3, 31 | txlm 21261 | . . . . 5 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉)) |
33 | 27, 28, 32 | mpbi2and 958 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉) |
34 | 33, 12 | lmcn 20919 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉))(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
35 | 26, 34 | eqbrtrrd 4607 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
36 | df-ov 6552 | . 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘〈𝑅, 𝑆〉) | |
37 | 35, 36 | syl6breqr 4625 | 1 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℤcz 11254 ℤ≥cuz 11563 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 ⇝𝑡clm 20840 ×t ctx 21173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-z 11255 df-uz 11564 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-cnp 20842 df-lm 20843 df-tx 21175 |
This theorem is referenced by: hlimadd 27434 |
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