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Theorem pcohtpy 24940

Description: Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
pcohtpy.4	⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
pcohtpy.5	⊢ (𝜑 → 𝐹( ≃_ph‘𝐽)𝐻)
pcohtpy.6	⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)

**Assertion**
Ref	Expression
pcohtpy	⊢ (𝜑 → (𝐹(_𝑝‘𝐽)𝐺)( ≃_ph‘𝐽)(𝐻(_𝑝‘𝐽)𝐾))

Proof of Theorem pcohtpy Dummy variables 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcohtpy.5	. . . . 5 ⊢ (𝜑 → 𝐹( ≃_ph‘𝐽)𝐻)
2		isphtpc 24913	. . . . 5 ⊢ (𝐹( ≃_ph‘𝐽)𝐻 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
3	1, 2	sylib 217	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
4	3	simp1d 1140	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		pcohtpy.6	. . . . 5 ⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)
6		isphtpc 24913	. . . . 5 ⊢ (𝐺( ≃_ph‘𝐽)𝐾 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
7	5, 6	sylib 217	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
8	7	simp1d 1140	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
9		pcohtpy.4	. . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
10	4, 8, 9	pcocn 24937	. 2 ⊢ (𝜑 → (𝐹(*_𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽))
11	3	simp2d 1141	. . 3 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
12	7	simp2d 1141	. . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐽))
13		phtpc01 24915	. . . . . 6 ⊢ (𝐹( ≃_ph‘𝐽)𝐻 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1)))
14	1, 13	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1)))
15	14	simprd 495	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐻‘1))
16		phtpc01 24915	. . . . . 6 ⊢ (𝐺( ≃_ph‘𝐽)𝐾 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1)))
17	5, 16	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1)))
18	17	simpld 494	. . . 4 ⊢ (𝜑 → (𝐺‘0) = (𝐾‘0))
19	9, 15, 18	3eqtr3d 2776	. . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0))
20	11, 12, 19	pcocn 24937	. 2 ⊢ (𝜑 → (𝐻(*_𝑝‘𝐽)𝐾) ∈ (II Cn 𝐽))
21	3	simp3d 1142	. . . . 5 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅)
22		n0 4342	. . . . 5 ⊢ ((𝐹(PHtpy‘𝐽)𝐻) ≠ ∅ ↔ ∃𝑚 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻))
23	21, 22	sylib 217	. . . 4 ⊢ (𝜑 → ∃𝑚 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻))
24	7	simp3d 1142	. . . . 5 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅)
25		n0 4342	. . . . 5 ⊢ ((𝐺(PHtpy‘𝐽)𝐾) ≠ ∅ ↔ ∃𝑛 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))
26	24, 25	sylib 217	. . . 4 ⊢ (𝜑 → ∃𝑛 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))
27		exdistrv 1952	. . . 4 ⊢ (∃𝑚∃𝑛(𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) ↔ (∃𝑚 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ ∃𝑛 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)))
28	23, 26, 27	sylanbrc 582	. . 3 ⊢ (𝜑 → ∃𝑚∃𝑛(𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)))
29	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → (𝐹‘1) = (𝐺‘0))
30	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝐹( ≃_ph‘𝐽)𝐻)
31	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝐺( ≃_ph‘𝐽)𝐾)
32		eqid 2728	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑚𝑦), (((2 · 𝑥) − 1)𝑛𝑦))) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑚𝑦), (((2 · 𝑥) − 1)𝑛𝑦)))
33		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻))
34		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))
35	29, 30, 31, 32, 33, 34	pcohtpylem 24939	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑚𝑦), (((2 · 𝑥) − 1)𝑛𝑦))) ∈ ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)))
36	35	ne0d 4331	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)) ≠ ∅)
37	36	ex 412	. . . 4 ⊢ (𝜑 → ((𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) → ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)) ≠ ∅))
38	37	exlimdvv 1930	. . 3 ⊢ (𝜑 → (∃𝑚∃𝑛(𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) → ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)) ≠ ∅))
39	28, 38	mpd 15	. 2 ⊢ (𝜑 → ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)) ≠ ∅)
40		isphtpc 24913	. 2 ⊢ ((𝐹(_𝑝‘𝐽)𝐺)( ≃_ph‘𝐽)(𝐻(_𝑝‘𝐽)𝐾) ↔ ((𝐹(_𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽) ∧ (𝐻(_𝑝‘𝐽)𝐾) ∈ (II Cn 𝐽) ∧ ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)) ≠ ∅))
41	10, 20, 39, 40	syl3anbrc 1341	1 ⊢ (𝜑 → (𝐹(_𝑝‘𝐽)𝐺)( ≃_ph‘𝐽)(𝐻(_𝑝‘𝐽)𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2936 ∅c0 4318 ifcif 4524 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 0cc0 11132 1c1 11133 · cmul 11137 ≤ cle 11273 − cmin 11468 / cdiv 11895 2c2 12291 [,]cicc 13353 Cn ccn 23121 IIcii 24788 PHtpycphtpy 24887 ≃_phcphtpc 24888 *_𝑝cpco 24920

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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-cn 23124 df-cnp 23125 df-tx 23459 df-hmeo 23652 df-xms 24219 df-ms 24220 df-tms 24221 df-ii 24790 df-htpy 24889 df-phtpy 24890 df-phtpc 24911 df-pco 24925

This theorem is referenced by: pcophtb 24949 pi1cpbl 24964 pi1xfrf 24973 pi1xfr 24975 pi1xfrcnvlem 24976

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