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Mirrors > Home > MPE Home > Th. List > isphtpy2d | Structured version Visualization version GIF version |
Description: Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
isphtpy.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
isphtpy.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
isphtpy2d.1 | ⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn 𝐽)) |
isphtpy2d.2 | ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘𝑠)) |
isphtpy2d.3 | ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐺‘𝑠)) |
isphtpy2d.4 | ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0)) |
isphtpy2d.5 | ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1)) |
Ref | Expression |
---|---|
isphtpy2d | ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isphtpy.2 | . 2 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
2 | isphtpy.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
3 | iitopon 23487 | . . . 4 ⊢ II ∈ (TopOn‘(0[,]1)) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
5 | isphtpy2d.1 | . . 3 ⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn 𝐽)) | |
6 | isphtpy2d.2 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘𝑠)) | |
7 | isphtpy2d.3 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐺‘𝑠)) | |
8 | 4, 1, 2, 5, 6, 7 | ishtpyd 23579 | . 2 ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) |
9 | isphtpy2d.4 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0)) | |
10 | isphtpy2d.5 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1)) | |
11 | 1, 2, 8, 9, 10 | isphtpyd 23590 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 [,]cicc 12742 TopOnctopon 21518 Cn ccn 21832 ×t ctx 22168 IIcii 23483 PHtpycphtpy 23572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-bases 21554 df-cn 21835 df-ii 23485 df-htpy 23574 df-phtpy 23575 |
This theorem is referenced by: reparphti 23601 pcohtpylem 23623 pcorevlem 23630 txsconnlem 32487 cvxsconn 32490 cvmliftphtlem 32564 |
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