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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 6882 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋) |
3 | txlm.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) | |
4 | 3 | ffvelrnda 6882 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌) |
5 | 2, 4 | opelxpd 5574 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 ∈ (𝑋 × 𝑌)) |
6 | eqidd 2737 | . . . . 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 22442 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
10 | 7, 8, 9 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
11 | lmcn2.o | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | |
12 | cntop2 22092 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁) → 𝑁 ∈ Top) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top) |
14 | toptopon2 21769 | . . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) | |
15 | 13, 14 | sylib 221 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
16 | cnf2 22100 | . . . . . . 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 1373 | . . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) |
18 | 17 | feqmptd 6758 | . . . . 5 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑥))) |
19 | fveq2 6695 | . . . . . 6 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
20 | df-ov 7194 | . . . . . 6 ⊢ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
21 | 19, 20 | eqtr4di 2789 | . . . . 5 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) |
22 | 5, 6, 18, 21 | fmptco 6922 | . . . 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 | eqtr4di 2789 | . . 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 2736 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
30 | 27, 28, 7, 8, 1, 3, 29 | txlm 22499 | . . . . 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 712 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉) |
32 | 31, 11 | lmcn 22156 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉))(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
33 | 24, 32 | eqbrtrrd 5063 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
34 | df-ov 7194 | . 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘〈𝑅, 𝑆〉) | |
35 | 33, 34 | breqtrrdi 5081 | 1 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 ∪ cuni 4805 class class class wbr 5039 ↦ cmpt 5120 × cxp 5534 ∘ ccom 5540 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℤcz 12141 ℤ≥cuz 12403 Topctop 21744 TopOnctopon 21761 Cn ccn 22075 ⇝𝑡clm 22077 ×t ctx 22411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-z 12142 df-uz 12404 df-topgen 16902 df-top 21745 df-topon 21762 df-bases 21797 df-cn 22078 df-cnp 22079 df-lm 22080 df-tx 22413 |
This theorem is referenced by: hlimadd 29228 |
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