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Mirrors > Home > MPE Home > Th. List > htpyi | Structured version Visualization version GIF version |
Description: A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.) |
Ref | Expression |
---|---|
ishtpy.1 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
ishtpy.3 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
ishtpy.4 | ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
htpyi.1 | ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) |
Ref | Expression |
---|---|
htpyi | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | htpyi.1 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) | |
2 | ishtpy.1 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | ishtpy.3 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
4 | ishtpy.4 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) | |
5 | 2, 3, 4 | ishtpy 25018 | . . . 4 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))) |
6 | 1, 5 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))) |
7 | 6 | simprd 495 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))) |
8 | oveq1 7438 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0)) | |
9 | fveq2 6907 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝐹‘𝑠) = (𝐹‘𝐴)) | |
10 | 8, 9 | eqeq12d 2751 | . . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹‘𝑠) ↔ (𝐴𝐻0) = (𝐹‘𝐴))) |
11 | oveq1 7438 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1)) | |
12 | fveq2 6907 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝐺‘𝑠) = (𝐺‘𝐴)) | |
13 | 11, 12 | eqeq12d 2751 | . . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺‘𝑠) ↔ (𝐴𝐻1) = (𝐺‘𝐴))) |
14 | 10, 13 | anbi12d 632 | . . 3 ⊢ (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ↔ ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))) |
15 | 14 | rspccva 3621 | . 2 ⊢ ((∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
16 | 7, 15 | sylan 580 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 TopOnctopon 22932 Cn ccn 23248 ×t ctx 23584 IIcii 24915 Htpy chtpy 25013 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-top 22916 df-topon 22933 df-cn 23251 df-htpy 25016 |
This theorem is referenced by: htpycom 25022 htpyco1 25024 htpyco2 25025 htpycc 25026 phtpy01 25031 pcohtpylem 25066 txsconnlem 35225 cvmliftphtlem 35302 |
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