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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 23539 | . . . 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 767 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽) | |
5 | simprr 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
6 | 4, 5 | oveq12d 7163 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌)) |
7 | pi1val.o | . . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
8 | 6, 7 | syl6eqr 2871 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂) |
9 | 4 | fveq2d 6667 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽)) |
10 | 8, 9 | oveq12d 7163 | . . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽))) |
11 | unieq 4838 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
12 | 11 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽) |
13 | pi1val.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
14 | toponuni 21450 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
16 | 15 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽) |
17 | 12, 16 | eqtr4d 2856 | . . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋) |
18 | topontop 21449 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 13, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | pi1val.2 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
21 | ovexd 7180 | . . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V) | |
22 | 3, 10, 17, 19, 20, 21 | ovmpodx 7290 | . 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽))) |
23 | 1, 22 | syl5eq 2865 | 1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cuni 4830 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 /s cqus 16766 Topctop 21429 TopOnctopon 21446 ≃phcphtpc 23500 Ω1 comi 23532 π1 cpi1 23534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-topon 21447 df-pi1 23539 |
This theorem is referenced by: pi1bas 23569 pi1addf 23578 pi1addval 23579 pi1grplem 23580 |
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