Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > phtpy01 | Structured version Visualization version GIF version |
Description: Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
isphtpy.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
isphtpy.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
phtpyi.1 | ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Ref | Expression |
---|---|
phtpy01 | ⊢ (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1elunit 12291 | . . . . 5 ⊢ 1 ∈ (0[,]1) | |
2 | isphtpy.2 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
3 | isphtpy.3 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
4 | phtpyi.1 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) | |
5 | 2, 3, 4 | phtpyi 22783 | . . . . 5 ⊢ ((𝜑 ∧ 1 ∈ (0[,]1)) → ((0𝐻1) = (𝐹‘0) ∧ (1𝐻1) = (𝐹‘1))) |
6 | 1, 5 | mpan2 707 | . . . 4 ⊢ (𝜑 → ((0𝐻1) = (𝐹‘0) ∧ (1𝐻1) = (𝐹‘1))) |
7 | 6 | simpld 475 | . . 3 ⊢ (𝜑 → (0𝐻1) = (𝐹‘0)) |
8 | 0elunit 12290 | . . . . 5 ⊢ 0 ∈ (0[,]1) | |
9 | iitopon 22682 | . . . . . . 7 ⊢ II ∈ (TopOn‘(0[,]1)) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
11 | 2, 3 | phtpyhtpy 22781 | . . . . . . 7 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺)) |
12 | 11, 4 | sseldd 3604 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) |
13 | 10, 2, 3, 12 | htpyi 22773 | . . . . 5 ⊢ ((𝜑 ∧ 0 ∈ (0[,]1)) → ((0𝐻0) = (𝐹‘0) ∧ (0𝐻1) = (𝐺‘0))) |
14 | 8, 13 | mpan2 707 | . . . 4 ⊢ (𝜑 → ((0𝐻0) = (𝐹‘0) ∧ (0𝐻1) = (𝐺‘0))) |
15 | 14 | simprd 479 | . . 3 ⊢ (𝜑 → (0𝐻1) = (𝐺‘0)) |
16 | 7, 15 | eqtr3d 2658 | . 2 ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) |
17 | 6 | simprd 479 | . . 3 ⊢ (𝜑 → (1𝐻1) = (𝐹‘1)) |
18 | 10, 2, 3, 12 | htpyi 22773 | . . . . 5 ⊢ ((𝜑 ∧ 1 ∈ (0[,]1)) → ((1𝐻0) = (𝐹‘1) ∧ (1𝐻1) = (𝐺‘1))) |
19 | 1, 18 | mpan2 707 | . . . 4 ⊢ (𝜑 → ((1𝐻0) = (𝐹‘1) ∧ (1𝐻1) = (𝐺‘1))) |
20 | 19 | simprd 479 | . . 3 ⊢ (𝜑 → (1𝐻1) = (𝐺‘1)) |
21 | 17, 20 | eqtr3d 2658 | . 2 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1)) |
22 | 16, 21 | jca 554 | 1 ⊢ (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 [,]cicc 12178 TopOnctopon 20715 Cn ccn 21028 IIcii 22678 Htpy chtpy 22766 PHtpycphtpy 22767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-ii 22680 df-htpy 22769 df-phtpy 22770 |
This theorem is referenced by: phtpycom 22787 phtpycc 22790 phtpc01 22796 pcohtpylem 22819 cvmliftphtlem 31299 |
Copyright terms: Public domain | W3C validator |