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Mirrors > Home > MPE Home > Th. List > isphtpyd | 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)) |
isphtpyd.1 | ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) |
isphtpyd.2 | ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0)) |
isphtpyd.3 | ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1)) |
Ref | Expression |
---|---|
isphtpyd | ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isphtpyd.1 | . 2 ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) | |
2 | isphtpyd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0)) | |
3 | isphtpyd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1)) | |
4 | 2, 3 | jca 514 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))) |
5 | 4 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))) |
6 | isphtpy.2 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
7 | isphtpy.3 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
8 | 6, 7 | isphtpy 23579 | . 2 ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))) |
9 | 1, 5, 8 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 [,]cicc 12735 Cn ccn 21826 IIcii 23477 Htpy chtpy 23565 PHtpycphtpy 23566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-top 21496 df-topon 21513 df-cn 21829 df-phtpy 23569 |
This theorem is referenced by: isphtpy2d 23585 phtpycom 23586 phtpyid 23587 phtpyco2 23588 phtpycc 23589 |
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