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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 23614 | . . . 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 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽) | |
5 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
6 | 4, 5 | oveq12d 7176 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌)) |
7 | pi1val.o | . . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
8 | 6, 7 | syl6eqr 2876 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂) |
9 | 4 | fveq2d 6676 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽)) |
10 | 8, 9 | oveq12d 7176 | . . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽))) |
11 | unieq 4851 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
12 | 11 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽) |
13 | pi1val.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
14 | toponuni 21524 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
16 | 15 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽) |
17 | 12, 16 | eqtr4d 2861 | . . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋) |
18 | topontop 21523 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
19 | 13, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
20 | pi1val.2 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
21 | ovexd 7193 | . . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V) | |
22 | 3, 10, 17, 19, 20, 21 | ovmpodx 7303 | . 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽))) |
23 | 1, 22 | syl5eq 2870 | 1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cuni 4840 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 /s cqus 16780 Topctop 21503 TopOnctopon 21520 ≃phcphtpc 23575 Ω1 comi 23607 π1 cpi1 23609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-topon 21521 df-pi1 23614 |
This theorem is referenced by: pi1bas 23644 pi1addf 23653 pi1addval 23654 pi1grplem 23655 |
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