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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 12848 | . . . . 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 23580 | . . . . 5 ⊢ ((𝜑 ∧ 1 ∈ (0[,]1)) → ((0𝐻1) = (𝐹‘0) ∧ (1𝐻1) = (𝐹‘1))) |
6 | 1, 5 | mpan2 689 | . . . 4 ⊢ (𝜑 → ((0𝐻1) = (𝐹‘0) ∧ (1𝐻1) = (𝐹‘1))) |
7 | 6 | simpld 497 | . . 3 ⊢ (𝜑 → (0𝐻1) = (𝐹‘0)) |
8 | 0elunit 12847 | . . . . 5 ⊢ 0 ∈ (0[,]1) | |
9 | iitopon 23479 | . . . . . . 7 ⊢ II ∈ (TopOn‘(0[,]1)) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
11 | 2, 3 | phtpyhtpy 23578 | . . . . . . 7 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺)) |
12 | 11, 4 | sseldd 3966 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) |
13 | 10, 2, 3, 12 | htpyi 23570 | . . . . 5 ⊢ ((𝜑 ∧ 0 ∈ (0[,]1)) → ((0𝐻0) = (𝐹‘0) ∧ (0𝐻1) = (𝐺‘0))) |
14 | 8, 13 | mpan2 689 | . . . 4 ⊢ (𝜑 → ((0𝐻0) = (𝐹‘0) ∧ (0𝐻1) = (𝐺‘0))) |
15 | 14 | simprd 498 | . . 3 ⊢ (𝜑 → (0𝐻1) = (𝐺‘0)) |
16 | 7, 15 | eqtr3d 2856 | . 2 ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) |
17 | 6 | simprd 498 | . . 3 ⊢ (𝜑 → (1𝐻1) = (𝐹‘1)) |
18 | 10, 2, 3, 12 | htpyi 23570 | . . . . 5 ⊢ ((𝜑 ∧ 1 ∈ (0[,]1)) → ((1𝐻0) = (𝐹‘1) ∧ (1𝐻1) = (𝐺‘1))) |
19 | 1, 18 | mpan2 689 | . . . 4 ⊢ (𝜑 → ((1𝐻0) = (𝐹‘1) ∧ (1𝐻1) = (𝐺‘1))) |
20 | 19 | simprd 498 | . . 3 ⊢ (𝜑 → (1𝐻1) = (𝐺‘1)) |
21 | 17, 20 | eqtr3d 2856 | . 2 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1)) |
22 | 16, 21 | jca 514 | 1 ⊢ (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 0cc0 10529 1c1 10530 [,]cicc 12733 TopOnctopon 21510 Cn ccn 21824 IIcii 23475 Htpy chtpy 23563 PHtpycphtpy 23564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-icc 12737 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-topgen 16709 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-top 21494 df-topon 21511 df-bases 21546 df-cn 21827 df-ii 23477 df-htpy 23566 df-phtpy 23567 |
This theorem is referenced by: phtpycom 23584 phtpycc 23587 phtpc01 23592 pcohtpylem 23615 cvmliftphtlem 32557 |
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