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Mirrors > Home > MPE Home > Th. List > htpyid | Structured version Visualization version GIF version |
Description: A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
htpyid.1 | ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) |
htpyid.2 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
htpyid.4 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Ref | Expression |
---|---|
htpyid | ⊢ (𝜑 → 𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | htpyid.2 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | htpyid.4 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
3 | htpyid.1 | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) | |
4 | iitopon 23414 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
6 | 1, 5 | cnmpt1st 22204 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
7 | 1, 5, 6, 2 | cnmpt21f 22208 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) ∈ ((𝐽 ×t II) Cn 𝐾)) |
8 | 3, 7 | eqeltrid 2914 | . 2 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 ×t II) Cn 𝐾)) |
9 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) | |
10 | 0elunit 12843 | . . 3 ⊢ 0 ∈ (0[,]1) | |
11 | fveq2 6663 | . . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠)) | |
12 | eqidd 2819 | . . . 4 ⊢ (𝑦 = 0 → (𝐹‘𝑠) = (𝐹‘𝑠)) | |
13 | fvex 6676 | . . . 4 ⊢ (𝐹‘𝑠) ∈ V | |
14 | 11, 12, 3, 13 | ovmpo 7299 | . . 3 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐹‘𝑠)) |
15 | 9, 10, 14 | sylancl 586 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐺0) = (𝐹‘𝑠)) |
16 | 1elunit 12844 | . . 3 ⊢ 1 ∈ (0[,]1) | |
17 | eqidd 2819 | . . . 4 ⊢ (𝑦 = 1 → (𝐹‘𝑠) = (𝐹‘𝑠)) | |
18 | 11, 17, 3, 13 | ovmpo 7299 | . . 3 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝐺1) = (𝐹‘𝑠)) |
19 | 9, 16, 18 | sylancl 586 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐺1) = (𝐹‘𝑠)) |
20 | 1, 2, 2, 8, 15, 19 | ishtpyd 23506 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 0cc0 10525 1c1 10526 [,]cicc 12729 TopOnctopon 21446 Cn ccn 21760 ×t ctx 22096 IIcii 23410 Htpy chtpy 23498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-cn 21763 df-tx 22098 df-ii 23412 df-htpy 23501 |
This theorem is referenced by: phtpyid 23520 |
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