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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 23090 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
6 | 1, 5 | cnmpt1st 21880 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
7 | 1, 5, 6, 2 | cnmpt21f 21884 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) ∈ ((𝐽 ×t II) Cn 𝐾)) |
8 | 3, 7 | syl5eqel 2862 | . 2 ⊢ (𝜑 → 𝐺 ∈ ((𝐽 ×t II) Cn 𝐾)) |
9 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) | |
10 | 0elunit 12605 | . . 3 ⊢ 0 ∈ (0[,]1) | |
11 | fveq2 6446 | . . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠)) | |
12 | eqidd 2778 | . . . 4 ⊢ (𝑦 = 0 → (𝐹‘𝑠) = (𝐹‘𝑠)) | |
13 | fvex 6459 | . . . 4 ⊢ (𝐹‘𝑠) ∈ V | |
14 | 11, 12, 3, 13 | ovmpt2 7073 | . . 3 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐹‘𝑠)) |
15 | 9, 10, 14 | sylancl 580 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐺0) = (𝐹‘𝑠)) |
16 | 1elunit 12606 | . . 3 ⊢ 1 ∈ (0[,]1) | |
17 | eqidd 2778 | . . . 4 ⊢ (𝑦 = 1 → (𝐹‘𝑠) = (𝐹‘𝑠)) | |
18 | 11, 17, 3, 13 | ovmpt2 7073 | . . 3 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝐺1) = (𝐹‘𝑠)) |
19 | 9, 16, 18 | sylancl 580 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐺1) = (𝐹‘𝑠)) |
20 | 1, 2, 2, 8, 15, 19 | ishtpyd 23182 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 0cc0 10272 1c1 10273 [,]cicc 12490 TopOnctopon 21122 Cn ccn 21436 ×t ctx 21772 IIcii 23086 Htpy chtpy 23174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-icc 12494 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-bases 21158 df-cn 21439 df-tx 21774 df-ii 23088 df-htpy 23177 |
This theorem is referenced by: phtpyid 23196 |
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