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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 12859 | . . . . 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 23591 | . . . . 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 12858 | . . . . 5 ⊢ 0 ∈ (0[,]1) | |
9 | iitopon 23490 | . . . . . . 7 ⊢ II ∈ (TopOn‘(0[,]1)) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
11 | 2, 3 | phtpyhtpy 23589 | . . . . . . 7 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺)) |
12 | 11, 4 | sseldd 3971 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) |
13 | 10, 2, 3, 12 | htpyi 23581 | . . . . 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 2861 | . 2 ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) |
17 | 6 | simprd 498 | . . 3 ⊢ (𝜑 → (1𝐻1) = (𝐹‘1)) |
18 | 10, 2, 3, 12 | htpyi 23581 | . . . . 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 2861 | . 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 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 [,]cicc 12744 TopOnctopon 21521 Cn ccn 21835 IIcii 23486 Htpy chtpy 23574 PHtpycphtpy 23575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-bases 21557 df-cn 21838 df-ii 23488 df-htpy 23577 df-phtpy 23578 |
This theorem is referenced by: phtpycom 23595 phtpycc 23598 phtpc01 23603 pcohtpylem 23626 cvmliftphtlem 32568 |
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