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Mirrors > Home > ILE Home > Th. List > lmcn2 | 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 5555 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋) |
3 | txlm.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) | |
4 | 3 | ffvelrnda 5555 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌) |
5 | 2, 4 | opelxpd 4572 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 ∈ (𝑋 × 𝑌)) |
6 | eqidd 2140 | . . . . 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 12434 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
10 | 7, 8, 9 | syl2anc 408 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
11 | lmcn2.o | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | |
12 | cntop2 12374 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁) → 𝑁 ∈ Top) | |
13 | 11, 12 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top) |
14 | toptopon2 12189 | . . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) | |
15 | 13, 14 | sylib 121 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
16 | cnf2 12377 | . . . . . . 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 1216 | . . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) |
18 | 17 | feqmptd 5474 | . . . . 5 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑥))) |
19 | fveq2 5421 | . . . . . 6 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
20 | df-ov 5777 | . . . . . 6 ⊢ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
21 | 19, 20 | syl6eqr 2190 | . . . . 5 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) |
22 | 5, 6, 18, 21 | fmptco 5586 | . . . 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 2190 | . . 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 2139 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
30 | 27, 28, 7, 8, 1, 3, 29 | txlm 12451 | . . . . 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 927 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉) |
32 | 31, 11 | lmcn 12423 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉))(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
33 | 24, 32 | eqbrtrrd 3952 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
34 | df-ov 5777 | . 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘〈𝑅, 𝑆〉) | |
35 | 33, 34 | breqtrrdi 3970 | 1 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 〈cop 3530 ∪ cuni 3736 class class class wbr 3929 ↦ cmpt 3989 × cxp 4537 ∘ ccom 4543 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ℤcz 9057 ℤ≥cuz 9329 Topctop 12167 TopOnctopon 12180 Cn ccn 12357 ⇝𝑡clm 12359 ×t ctx 12424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pm 6545 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-topgen 12144 df-top 12168 df-topon 12181 df-bases 12213 df-cn 12360 df-cnp 12361 df-lm 12362 df-tx 12425 |
This theorem is referenced by: (None) |
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