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Mirrors > Home > MPE Home > Th. List > phtpcco2 | Structured version Visualization version GIF version |
Description: Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
phtpcco2.f | ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺) |
phtpcco2.p | ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) |
Ref | Expression |
---|---|
phtpcco2 | ⊢ (𝜑 → (𝑃 ∘ 𝐹)( ≃ph‘𝐾)(𝑃 ∘ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phtpcco2.f | . . . . 5 ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺) | |
2 | isphtpc 23845 | . . . . 5 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅)) | |
3 | 1, 2 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅)) |
4 | 3 | simp1d 1144 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
5 | phtpcco2.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) | |
6 | cnco 22117 | . . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → (𝑃 ∘ 𝐹) ∈ (II Cn 𝐾)) | |
7 | 4, 5, 6 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑃 ∘ 𝐹) ∈ (II Cn 𝐾)) |
8 | 3 | simp2d 1145 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
9 | cnco 22117 | . . 3 ⊢ ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → (𝑃 ∘ 𝐺) ∈ (II Cn 𝐾)) | |
10 | 8, 5, 9 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑃 ∘ 𝐺) ∈ (II Cn 𝐾)) |
11 | 3 | simp3d 1146 | . . . 4 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) |
12 | n0 4247 | . . . 4 ⊢ ((𝐹(PHtpy‘𝐽)𝐺) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) | |
13 | 11, 12 | sylib 221 | . . 3 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
14 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) → 𝐹 ∈ (II Cn 𝐽)) |
15 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) → 𝐺 ∈ (II Cn 𝐽)) |
16 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) → 𝑃 ∈ (𝐽 Cn 𝐾)) |
17 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) → 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) | |
18 | 14, 15, 16, 17 | phtpyco2 23841 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) → (𝑃 ∘ 𝑓) ∈ ((𝑃 ∘ 𝐹)(PHtpy‘𝐾)(𝑃 ∘ 𝐺))) |
19 | 18 | ne0d 4236 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐹(PHtpy‘𝐽)𝐺)) → ((𝑃 ∘ 𝐹)(PHtpy‘𝐾)(𝑃 ∘ 𝐺)) ≠ ∅) |
20 | 13, 19 | exlimddv 1943 | . 2 ⊢ (𝜑 → ((𝑃 ∘ 𝐹)(PHtpy‘𝐾)(𝑃 ∘ 𝐺)) ≠ ∅) |
21 | isphtpc 23845 | . 2 ⊢ ((𝑃 ∘ 𝐹)( ≃ph‘𝐾)(𝑃 ∘ 𝐺) ↔ ((𝑃 ∘ 𝐹) ∈ (II Cn 𝐾) ∧ (𝑃 ∘ 𝐺) ∈ (II Cn 𝐾) ∧ ((𝑃 ∘ 𝐹)(PHtpy‘𝐾)(𝑃 ∘ 𝐺)) ≠ ∅)) | |
22 | 7, 10, 20, 21 | syl3anbrc 1345 | 1 ⊢ (𝜑 → (𝑃 ∘ 𝐹)( ≃ph‘𝐾)(𝑃 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 class class class wbr 5039 ∘ ccom 5540 ‘cfv 6358 (class class class)co 7191 Cn ccn 22075 IIcii 23726 PHtpycphtpy 23819 ≃phcphtpc 23820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-icc 12907 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 df-cn 22078 df-tx 22413 df-ii 23728 df-htpy 23821 df-phtpy 23822 df-phtpc 23843 |
This theorem is referenced by: pi1cof 23910 cvmlift3lem1 32948 |
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