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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 23214 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
5 | 2, 3, 4 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
6 | htpyco1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿)) | |
7 | cnco 23214 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
8 | 2, 6, 7 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
9 | htpyco1.n | . . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) | |
10 | iitopon 24843 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
12 | 1, 11 | cnmpt1st 23616 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
13 | 1, 11, 12, 2 | cnmpt21f 23620 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×t II) Cn 𝐾)) |
14 | 1, 11 | cnmpt2nd 23617 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×t II) Cn II)) |
15 | cntop2 23189 | . . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
16 | 2, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
17 | toptopon2 22864 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
18 | 16, 17 | sylib 217 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
19 | 18, 3, 6 | htpycn 24943 | . . . . 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 3977 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×t II) Cn 𝐿)) |
22 | 1, 11, 13, 14, 21 | cnmpt22f 23623 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×t II) Cn 𝐿)) |
23 | 9, 22 | eqeltrid 2829 | . 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×t II) Cn 𝐿)) |
24 | cnf2 23197 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾) | |
25 | 1, 18, 2, 24 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾) |
26 | 25 | ffvelcdmda 7093 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾) |
27 | 18, 3, 6, 20 | htpyi 24944 | . . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
28 | 26, 27 | syldan 589 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
29 | 28 | simpld 493 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠))) |
30 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) | |
31 | 0elunit 13481 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
32 | fveq2 6896 | . . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠)) | |
33 | id 22 | . . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0) | |
34 | 32, 33 | oveqan12d 7438 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0)) |
35 | ovex 7452 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V | |
36 | 34, 9, 35 | ovmpoa 7576 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
37 | 30, 31, 36 | sylancl 584 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
38 | fvco3 6996 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) | |
39 | 25, 38 | sylan 578 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) |
40 | 29, 37, 39 | 3eqtr4d 2775 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠)) |
41 | 28 | simprd 494 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))) |
42 | 1elunit 13482 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
43 | id 22 | . . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1) | |
44 | 32, 43 | oveqan12d 7438 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1)) |
45 | ovex 7452 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V | |
46 | 44, 9, 45 | ovmpoa 7576 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
47 | 30, 42, 46 | sylancl 584 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
48 | fvco3 6996 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) | |
49 | 25, 48 | sylan 578 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) |
50 | 41, 47, 49 | 3eqtr4d 2775 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠)) |
51 | 1, 5, 8, 23, 40, 50 | ishtpyd 24945 | 1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cuni 4909 ∘ ccom 5682 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 0cc0 11140 1c1 11141 [,]cicc 13362 Topctop 22839 TopOnctopon 22856 Cn ccn 23172 ×t ctx 23508 IIcii 24839 Htpy chtpy 24937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-icc 13366 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22840 df-topon 22857 df-bases 22893 df-cn 23175 df-tx 23510 df-ii 24841 df-htpy 24940 |
This theorem is referenced by: (None) |
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