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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 21568 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
5 | 2, 3, 4 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
6 | htpyco1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿)) | |
7 | cnco 21568 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
8 | 2, 6, 7 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
9 | htpyco1.n | . . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) | |
10 | iitopon 23180 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
12 | 1, 11 | cnmpt1st 21970 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
13 | 1, 11, 12, 2 | cnmpt21f 21974 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×t II) Cn 𝐾)) |
14 | 1, 11 | cnmpt2nd 21971 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×t II) Cn II)) |
15 | cntop2 21543 | . . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
16 | 2, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
17 | toptopon2 21220 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
18 | 16, 17 | sylib 210 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
19 | 18, 3, 6 | htpycn 23270 | . . . . 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 3855 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×t II) Cn 𝐿)) |
22 | 1, 11, 13, 14, 21 | cnmpt22f 21977 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×t II) Cn 𝐿)) |
23 | 9, 22 | syl5eqel 2864 | . 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×t II) Cn 𝐿)) |
24 | cnf2 21551 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾) | |
25 | 1, 18, 2, 24 | syl3anc 1351 | . . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾) |
26 | 25 | ffvelrnda 6670 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾) |
27 | 18, 3, 6, 20 | htpyi 23271 | . . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
28 | 26, 27 | syldan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
29 | 28 | simpld 487 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠))) |
30 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) | |
31 | 0elunit 12664 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
32 | fveq2 6493 | . . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠)) | |
33 | id 22 | . . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0) | |
34 | 32, 33 | oveqan12d 6989 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0)) |
35 | ovex 7002 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V | |
36 | 34, 9, 35 | ovmpoa 7115 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
37 | 30, 31, 36 | sylancl 577 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
38 | fvco3 6582 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) | |
39 | 25, 38 | sylan 572 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) |
40 | 29, 37, 39 | 3eqtr4d 2818 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠)) |
41 | 28 | simprd 488 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))) |
42 | 1elunit 12665 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
43 | id 22 | . . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1) | |
44 | 32, 43 | oveqan12d 6989 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1)) |
45 | ovex 7002 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V | |
46 | 44, 9, 45 | ovmpoa 7115 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
47 | 30, 42, 46 | sylancl 577 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
48 | fvco3 6582 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) | |
49 | 25, 48 | sylan 572 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) |
50 | 41, 47, 49 | 3eqtr4d 2818 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠)) |
51 | 1, 5, 8, 23, 40, 50 | ishtpyd 23272 | 1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∪ cuni 4706 ∘ ccom 5404 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 0cc0 10327 1c1 10328 [,]cicc 12550 Topctop 21195 TopOnctopon 21212 Cn ccn 21526 ×t ctx 21862 IIcii 23176 Htpy chtpy 23264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-icc 12554 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-topgen 16563 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-bases 21248 df-cn 21529 df-tx 21864 df-ii 23178 df-htpy 23267 |
This theorem is referenced by: (None) |
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