Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > htpyco1 | Structured version Visualization version GIF version |
Description: Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
htpyco1.n | ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) |
htpyco1.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
htpyco1.p | ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) |
htpyco1.f | ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) |
htpyco1.g | ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿)) |
htpyco1.h | ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺)) |
Ref | Expression |
---|---|
htpyco1 | ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | htpyco1.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | htpyco1.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) | |
3 | htpyco1.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) | |
4 | cnco 22117 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
5 | 2, 3, 4 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
6 | htpyco1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿)) | |
7 | cnco 22117 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
8 | 2, 6, 7 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
9 | htpyco1.n | . . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) | |
10 | iitopon 23730 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
12 | 1, 11 | cnmpt1st 22519 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
13 | 1, 11, 12, 2 | cnmpt21f 22523 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×t II) Cn 𝐾)) |
14 | 1, 11 | cnmpt2nd 22520 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×t II) Cn II)) |
15 | cntop2 22092 | . . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
16 | 2, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
17 | toptopon2 21769 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
18 | 16, 17 | sylib 221 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
19 | 18, 3, 6 | htpycn 23824 | . . . . 5 ⊢ (𝜑 → (𝐹(𝐾 Htpy 𝐿)𝐺) ⊆ ((𝐾 ×t II) Cn 𝐿)) |
20 | htpyco1.h | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺)) | |
21 | 19, 20 | sseldd 3888 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×t II) Cn 𝐿)) |
22 | 1, 11, 13, 14, 21 | cnmpt22f 22526 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×t II) Cn 𝐿)) |
23 | 9, 22 | eqeltrid 2835 | . 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×t II) Cn 𝐿)) |
24 | cnf2 22100 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾) | |
25 | 1, 18, 2, 24 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾) |
26 | 25 | ffvelrnda 6882 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾) |
27 | 18, 3, 6, 20 | htpyi 23825 | . . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
28 | 26, 27 | syldan 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
29 | 28 | simpld 498 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠))) |
30 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) | |
31 | 0elunit 13022 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
32 | fveq2 6695 | . . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠)) | |
33 | id 22 | . . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0) | |
34 | 32, 33 | oveqan12d 7210 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0)) |
35 | ovex 7224 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V | |
36 | 34, 9, 35 | ovmpoa 7342 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
37 | 30, 31, 36 | sylancl 589 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
38 | fvco3 6788 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) | |
39 | 25, 38 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) |
40 | 29, 37, 39 | 3eqtr4d 2781 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠)) |
41 | 28 | simprd 499 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))) |
42 | 1elunit 13023 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
43 | id 22 | . . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1) | |
44 | 32, 43 | oveqan12d 7210 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1)) |
45 | ovex 7224 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V | |
46 | 44, 9, 45 | ovmpoa 7342 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
47 | 30, 42, 46 | sylancl 589 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
48 | fvco3 6788 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) | |
49 | 25, 48 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) |
50 | 41, 47, 49 | 3eqtr4d 2781 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠)) |
51 | 1, 5, 8, 23, 40, 50 | ishtpyd 23826 | 1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∪ cuni 4805 ∘ ccom 5540 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 0cc0 10694 1c1 10695 [,]cicc 12903 Topctop 21744 TopOnctopon 21761 Cn ccn 22075 ×t ctx 22411 IIcii 23726 Htpy chtpy 23818 |
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 ax-pre-sup 10772 |
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-rmo 3059 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-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-icc 12907 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 df-cn 22078 df-tx 22413 df-ii 23728 df-htpy 23821 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |