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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 23882 | . . . 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 235 | . . 3 ⊢ (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))) |
7 | 6 | simprd 499 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))) |
8 | oveq1 7229 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0)) | |
9 | fveq2 6726 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝐹‘𝑠) = (𝐹‘𝐴)) | |
10 | 8, 9 | eqeq12d 2754 | . . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹‘𝑠) ↔ (𝐴𝐻0) = (𝐹‘𝐴))) |
11 | oveq1 7229 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1)) | |
12 | fveq2 6726 | . . . . 5 ⊢ (𝑠 = 𝐴 → (𝐺‘𝑠) = (𝐺‘𝐴)) | |
13 | 11, 12 | eqeq12d 2754 | . . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺‘𝑠) ↔ (𝐴𝐻1) = (𝐺‘𝐴))) |
14 | 10, 13 | anbi12d 634 | . . 3 ⊢ (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ↔ ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))) |
15 | 14 | rspccva 3543 | . 2 ⊢ ((∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
16 | 7, 15 | sylan 583 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ‘cfv 6389 (class class class)co 7222 0cc0 10742 1c1 10743 TopOnctopon 21820 Cn ccn 22134 ×t ctx 22470 IIcii 23785 Htpy chtpy 23877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-fv 6397 df-ov 7225 df-oprab 7226 df-mpo 7227 df-1st 7770 df-2nd 7771 df-map 8519 df-top 21804 df-topon 21821 df-cn 22137 df-htpy 23880 |
This theorem is referenced by: htpycom 23886 htpyco1 23888 htpyco2 23889 htpycc 23890 phtpy01 23895 pcohtpylem 23929 txsconnlem 32928 cvmliftphtlem 33005 |
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