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Mirrors > Home > MPE Home > Th. List > pi1val | Structured version Visualization version GIF version |
Description: The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
pi1val.g | ⊢ 𝐺 = (𝐽 π1 𝑌) |
pi1val.1 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
pi1val.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
pi1val.o | ⊢ 𝑂 = (𝐽 Ω1 𝑌) |
Ref | Expression |
---|---|
pi1val | ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1val.g | . 2 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
2 | df-pi1 22616 | . . . 4 ⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗))) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))) |
4 | simprl 790 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽) | |
5 | simprr 792 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
6 | 4, 5 | oveq12d 6567 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌)) |
7 | pi1val.o | . . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
8 | 6, 7 | syl6eqr 2662 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂) |
9 | 4 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽)) |
10 | 8, 9 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽))) |
11 | unieq 4380 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽) |
13 | pi1val.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
14 | toponuni 20542 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
16 | 15 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽) |
17 | 12, 16 | eqtr4d 2647 | . . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋) |
18 | topontop 20541 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 13, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | pi1val.2 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
21 | ovex 6577 | . . . 4 ⊢ (𝑂 /s ( ≃ph‘𝐽)) ∈ V | |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V) |
23 | 3, 10, 17, 19, 20, 22 | ovmpt2dx 6685 | . 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽))) |
24 | 1, 23 | syl5eq 2656 | 1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 /s cqus 15988 Topctop 20517 TopOnctopon 20518 ≃phcphtpc 22576 Ω1 comi 22609 π1 cpi1 22611 |
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-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-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-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-topon 20523 df-pi1 22616 |
This theorem is referenced by: pi1bas 22646 pi1addf 22655 pi1addval 22656 pi1grplem 22657 |
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