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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 21874 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
5 | 2, 3, 4 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
6 | htpyco1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿)) | |
7 | cnco 21874 | . . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) | |
8 | 2, 6, 7 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿)) |
9 | htpyco1.n | . . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) | |
10 | iitopon 23487 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
12 | 1, 11 | cnmpt1st 22276 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
13 | 1, 11, 12, 2 | cnmpt21f 22280 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×t II) Cn 𝐾)) |
14 | 1, 11 | cnmpt2nd 22277 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×t II) Cn II)) |
15 | cntop2 21849 | . . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
16 | 2, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top) |
17 | toptopon2 21526 | . . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | |
18 | 16, 17 | sylib 220 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
19 | 18, 3, 6 | htpycn 23577 | . . . . 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 3968 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×t II) Cn 𝐿)) |
22 | 1, 11, 13, 14, 21 | cnmpt22f 22283 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×t II) Cn 𝐿)) |
23 | 9, 22 | eqeltrid 2917 | . 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×t II) Cn 𝐿)) |
24 | cnf2 21857 | . . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾) | |
25 | 1, 18, 2, 24 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾) |
26 | 25 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾) |
27 | 18, 3, 6, 20 | htpyi 23578 | . . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
28 | 26, 27 | syldan 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))) |
29 | 28 | simpld 497 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠))) |
30 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) | |
31 | 0elunit 12856 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
32 | fveq2 6670 | . . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠)) | |
33 | id 22 | . . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0) | |
34 | 32, 33 | oveqan12d 7175 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0)) |
35 | ovex 7189 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V | |
36 | 34, 9, 35 | ovmpoa 7305 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
37 | 30, 31, 36 | sylancl 588 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0)) |
38 | fvco3 6760 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) | |
39 | 25, 38 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠))) |
40 | 29, 37, 39 | 3eqtr4d 2866 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠)) |
41 | 28 | simprd 498 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))) |
42 | 1elunit 12857 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
43 | id 22 | . . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1) | |
44 | 32, 43 | oveqan12d 7175 | . . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1)) |
45 | ovex 7189 | . . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V | |
46 | 44, 9, 45 | ovmpoa 7305 | . . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
47 | 30, 42, 46 | sylancl 588 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1)) |
48 | fvco3 6760 | . . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) | |
49 | 25, 48 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠))) |
50 | 41, 47, 49 | 3eqtr4d 2866 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠)) |
51 | 1, 5, 8, 23, 40, 50 | ishtpyd 23579 | 1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cuni 4838 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 0cc0 10537 1c1 10538 [,]cicc 12742 Topctop 21501 TopOnctopon 21518 Cn ccn 21832 ×t ctx 22168 IIcii 23483 Htpy chtpy 23571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-bases 21554 df-cn 21835 df-tx 22170 df-ii 23485 df-htpy 23574 |
This theorem is referenced by: (None) |
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