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Mirrors > Home > MPE Home > Th. List > cncfcompt2 | Structured version Visualization version GIF version |
Description: Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
cncfcompt2.xph | ⊢ Ⅎ𝑥𝜑 |
cncfcompt2.ab | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵)) |
cncfcompt2.cd | ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸)) |
cncfcompt2.bc | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
cncfcompt2.st | ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) |
Ref | Expression |
---|---|
cncfcompt2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfcompt2.xph | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
2 | cncfcompt2.bc | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | 2 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
4 | cncfcompt2.ab | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵)) | |
5 | cncff 24901 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵) | |
6 | 4, 5 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵) |
7 | 6 | fvmptelcdm 7119 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵) |
8 | 3, 7 | sseldd 3979 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐶) |
9 | 8 | ex 411 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑅 ∈ 𝐶)) |
10 | 1, 9 | ralrimi 3245 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐶) |
11 | eqidd 2727 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)) | |
12 | eqidd 2727 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) = (𝑦 ∈ 𝐶 ↦ 𝑆)) | |
13 | cncfcompt2.st | . . . 4 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) | |
14 | 10, 11, 12, 13 | fmptcof 7136 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
15 | 14 | eqcomd 2732 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) = ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅))) |
16 | cncfcompt2.cd | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸)) | |
17 | cncfrss 24899 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸) → 𝐶 ⊆ ℂ) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ ℂ) |
19 | cncfss 24907 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) | |
20 | 2, 18, 19 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) |
21 | 20, 4 | sseldd 3979 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐶)) |
22 | 21, 16 | cncfco 24915 | . 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐶 ↦ 𝑆) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) ∈ (𝐴–cn→𝐸)) |
23 | 15, 22 | eqeltrd 2826 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ⊆ wss 3946 ↦ cmpt 5228 ∘ ccom 5678 ⟶wf 6542 (class class class)co 7416 ℂcc 11147 –cn→ccncf 24884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-2 12321 df-cj 15099 df-re 15100 df-im 15101 df-abs 15236 df-cncf 24886 |
This theorem is referenced by: lcmineqlem9 41749 lcmineqlem12 41752 etransclem18 45909 etransclem22 45913 etransclem46 45937 |
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