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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 22579 | . . . 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 221 | . . 3 ⊢ (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))) |
7 | 6 | simprd 478 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))) |
8 | oveq1 6556 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0)) | |
9 | fveq2 6103 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝐹‘𝑠) = (𝐹‘𝐴)) | |
10 | 8, 9 | eqeq12d 2625 | . . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹‘𝑠) ↔ (𝐴𝐻0) = (𝐹‘𝐴))) |
11 | oveq1 6556 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1)) | |
12 | fveq2 6103 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝐺‘𝑠) = (𝐺‘𝐴)) | |
13 | 11, 12 | eqeq12d 2625 | . . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺‘𝑠) ↔ (𝐴𝐻1) = (𝐺‘𝐴))) |
14 | 10, 13 | anbi12d 743 | . . 3 ⊢ (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ↔ ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))) |
15 | 14 | rspccva 3281 | . 2 ⊢ ((∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
16 | 7, 15 | sylan 487 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 TopOnctopon 20518 Cn ccn 20838 ×t ctx 21173 IIcii 22486 Htpy chtpy 22574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-top 20521 df-topon 20523 df-cn 20841 df-htpy 22577 |
This theorem is referenced by: htpycom 22583 htpyco1 22585 htpyco2 22586 htpycc 22587 phtpy01 22592 pcohtpylem 22627 txsconlem 30476 cvmliftphtlem 30553 |
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