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Mirrors > Home > MPE Home > Th. List > phtpyid | Structured version Visualization version GIF version |
Description: A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
phtpyid.1 | ⊢ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) |
phtpyid.3 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
Ref | Expression |
---|---|
phtpyid | ⊢ (𝜑 → 𝐺 ∈ (𝐹(PHtpy‘𝐽)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phtpyid.3 | . 2 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
2 | phtpyid.1 | . . 3 ⊢ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) | |
3 | iitopon 23484 | . . . 4 ⊢ II ∈ (TopOn‘(0[,]1)) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
5 | 2, 4, 1 | htpyid 23582 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝐹(II Htpy 𝐽)𝐹)) |
6 | 0elunit 12847 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
7 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0)) | |
8 | eqidd 2799 | . . . . 5 ⊢ (𝑦 = 𝑠 → (𝐹‘0) = (𝐹‘0)) | |
9 | fvex 6658 | . . . . 5 ⊢ (𝐹‘0) ∈ V | |
10 | 7, 8, 2, 9 | ovmpo 7289 | . . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐹‘0)) |
11 | 6, 10 | mpan 689 | . . 3 ⊢ (𝑠 ∈ (0[,]1) → (0𝐺𝑠) = (𝐹‘0)) |
12 | 11 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐹‘0)) |
13 | 1elunit 12848 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
14 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1)) | |
15 | eqidd 2799 | . . . . 5 ⊢ (𝑦 = 𝑠 → (𝐹‘1) = (𝐹‘1)) | |
16 | fvex 6658 | . . . . 5 ⊢ (𝐹‘1) ∈ V | |
17 | 14, 15, 2, 16 | ovmpo 7289 | . . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐹‘1)) |
18 | 13, 17 | mpan 689 | . . 3 ⊢ (𝑠 ∈ (0[,]1) → (1𝐺𝑠) = (𝐹‘1)) |
19 | 18 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐹‘1)) |
20 | 1, 1, 5, 12, 19 | isphtpyd 23591 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐹(PHtpy‘𝐽)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 1c1 10527 [,]cicc 12729 TopOnctopon 21515 Cn ccn 21829 IIcii 23480 PHtpycphtpy 23573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 df-tx 22167 df-ii 23482 df-htpy 23575 df-phtpy 23576 |
This theorem is referenced by: phtpcer 23600 |
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