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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 6678 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋) |
3 | txlm.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) | |
4 | 3 | ffvelrnda 6678 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌) |
5 | 2, 4 | opelxpd 5446 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 ∈ (𝑋 × 𝑌)) |
6 | eqidd 2779 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
7 | txlm.j | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
8 | txlm.k | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
9 | txtopon 21906 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
10 | 7, 8, 9 | syl2anc 576 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
11 | lmcn2.o | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | |
12 | cntop2 21556 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁) → 𝑁 ∈ Top) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top) |
14 | toptopon2 21233 | . . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) | |
15 | 13, 14 | sylib 210 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
16 | cnf2 21564 | . . . . . . 7 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) | |
17 | 10, 15, 11, 16 | syl3anc 1351 | . . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) |
18 | 17 | feqmptd 6564 | . . . . 5 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑥))) |
19 | fveq2 6501 | . . . . . 6 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
20 | df-ov 6981 | . . . . . 6 ⊢ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
21 | 19, 20 | syl6eqr 2832 | . . . . 5 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) |
22 | 5, 6, 18, 21 | fmptco 6716 | . . . 4 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)))) |
23 | lmcn2.h | . . . 4 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) | |
24 | 22, 23 | syl6eqr 2832 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) = 𝐻) |
25 | lmcn2.fl | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑅) | |
26 | lmcn2.gl | . . . . 5 ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐾)𝑆) | |
27 | txlm.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
28 | txlm.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
29 | eqid 2778 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
30 | 27, 28, 7, 8, 1, 3, 29 | txlm 21963 | . . . . 5 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉)) |
31 | 25, 26, 30 | mpbi2and 699 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉) |
32 | 31, 11 | lmcn 21620 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉))(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
33 | 24, 32 | eqbrtrrd 4954 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
34 | df-ov 6981 | . 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘〈𝑅, 𝑆〉) | |
35 | 33, 34 | syl6breqr 4972 | 1 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 〈cop 4448 ∪ cuni 4713 class class class wbr 4930 ↦ cmpt 5009 × cxp 5406 ∘ ccom 5412 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ℤcz 11796 ℤ≥cuz 12061 Topctop 21208 TopOnctopon 21225 Cn ccn 21539 ⇝𝑡clm 21541 ×t ctx 21875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-map 8210 df-pm 8211 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-z 11797 df-uz 12062 df-topgen 16576 df-top 21209 df-topon 21226 df-bases 21261 df-cn 21542 df-cnp 21543 df-lm 21544 df-tx 21877 |
This theorem is referenced by: hlimadd 28752 |
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