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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 | ffvelcdmda 7028 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋) |
| 3 | txlm.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) | |
| 4 | 3 | ffvelcdmda 7028 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌) |
| 5 | 2, 4 | opelxpd 5661 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 ∈ (𝑋 × 𝑌)) |
| 6 | eqidd 2738 | . . . . 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 23534 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
| 10 | 7, 8, 9 | syl2anc 585 | . . . . . . 7 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
| 11 | lmcn2.o | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | |
| 12 | cntop2 23184 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁) → 𝑁 ∈ Top) | |
| 13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top) |
| 14 | toptopon2 22861 | . . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁)) | |
| 15 | 13, 14 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁)) |
| 16 | cnf2 23192 | . . . . . . 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 1374 | . . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁) |
| 18 | 17 | feqmptd 6900 | . . . . 5 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑥))) |
| 19 | fveq2 6832 | . . . . . 6 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉)) | |
| 20 | df-ov 7361 | . . . . . 6 ⊢ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)) = (𝑂‘〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
| 21 | 19, 20 | eqtr4di 2790 | . . . . 5 ⊢ (𝑥 = 〈(𝐹‘𝑛), (𝐺‘𝑛)〉 → (𝑂‘𝑥) = ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) |
| 22 | 5, 6, 18, 21 | fmptco 7074 | . . . 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 2790 | . . 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 2737 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) | |
| 30 | 27, 28, 7, 8, 1, 3, 29 | txlm 23591 | . . . . 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 713 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉)(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉) |
| 32 | 31, 11 | lmcn 23248 | . . 3 ⊢ (𝜑 → (𝑂 ∘ (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉))(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
| 33 | 24, 32 | eqbrtrrd 5110 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑂‘〈𝑅, 𝑆〉)) |
| 34 | df-ov 7361 | . 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘〈𝑅, 𝑆〉) | |
| 35 | 33, 34 | breqtrrdi 5128 | 1 ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 〈cop 4574 ∪ cuni 4851 class class class wbr 5086 ↦ cmpt 5167 × cxp 5620 ∘ ccom 5626 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ℤcz 12489 ℤ≥cuz 12752 Topctop 22836 TopOnctopon 22853 Cn ccn 23167 ⇝𝑡clm 23169 ×t ctx 23503 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-map 8766 df-pm 8767 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-z 12490 df-uz 12753 df-topgen 17364 df-top 22837 df-topon 22854 df-bases 22889 df-cn 23170 df-cnp 23171 df-lm 23172 df-tx 23505 |
| This theorem is referenced by: hlimadd 31253 |
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