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Mirrors > Home > MPE Home > Th. List > phtpyi | Structured version Visualization version GIF version |
Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
isphtpy.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
isphtpy.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
phtpyi.1 | ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Ref | Expression |
---|---|
phtpyi | ⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phtpyi.1 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) | |
2 | isphtpy.2 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
3 | isphtpy.3 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
4 | 2, 3 | isphtpy 22588 | . . . 4 ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))) |
5 | 1, 4 | mpbid 221 | . . 3 ⊢ (𝜑 → (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))) |
6 | 5 | simprd 478 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))) |
7 | oveq2 6557 | . . . . 5 ⊢ (𝑠 = 𝐴 → (0𝐻𝑠) = (0𝐻𝐴)) | |
8 | 7 | eqeq1d 2612 | . . . 4 ⊢ (𝑠 = 𝐴 → ((0𝐻𝑠) = (𝐹‘0) ↔ (0𝐻𝐴) = (𝐹‘0))) |
9 | oveq2 6557 | . . . . 5 ⊢ (𝑠 = 𝐴 → (1𝐻𝑠) = (1𝐻𝐴)) | |
10 | 9 | eqeq1d 2612 | . . . 4 ⊢ (𝑠 = 𝐴 → ((1𝐻𝑠) = (𝐹‘1) ↔ (1𝐻𝐴) = (𝐹‘1))) |
11 | 8, 10 | anbi12d 743 | . . 3 ⊢ (𝑠 = 𝐴 → (((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ↔ ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))) |
12 | 11 | rspccva 3281 | . 2 ⊢ ((∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))) |
13 | 6, 12 | sylan 487 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))) |
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 [,]cicc 12049 Cn ccn 20838 IIcii 22486 Htpy chtpy 22574 PHtpycphtpy 22575 |
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-rep 4699 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-reu 2903 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-f1 5809 df-fo 5810 df-f1o 5811 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-phtpy 22578 |
This theorem is referenced by: phtpy01 22592 phtpycom 22595 phtpyco2 22597 phtpycc 22598 pcohtpylem 22627 txsconlem 30476 cvmliftphtlem 30553 cvmliftpht 30554 |
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