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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 25041 | . . . 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 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽) | |
| 5 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
| 6 | 4, 5 | oveq12d 7449 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = (𝐽 Ω1 𝑌)) |
| 7 | pi1val.o | . . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 Ω1 𝑦) = 𝑂) |
| 9 | 4 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ( ≃ph‘𝑗) = ( ≃ph‘𝐽)) |
| 10 | 8, 9 | oveq12d 7449 | . . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) = (𝑂 /s ( ≃ph‘𝐽))) |
| 11 | unieq 4918 | . . . . 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 22920 | . . . . . 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 2780 | . . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋) |
| 18 | topontop 22919 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
| 19 | 13, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
| 20 | pi1val.2 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 21 | ovexd 7466 | . . 3 ⊢ (𝜑 → (𝑂 /s ( ≃ph‘𝐽)) ∈ V) | |
| 22 | 3, 10, 17, 19, 20, 21 | ovmpodx 7584 | . 2 ⊢ (𝜑 → (𝐽 π1 𝑌) = (𝑂 /s ( ≃ph‘𝐽))) |
| 23 | 1, 22 | eqtrid 2789 | 1 ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 /s cqus 17550 Topctop 22899 TopOnctopon 22916 ≃phcphtpc 25001 Ω1 comi 25034 π1 cpi1 25036 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-topon 22917 df-pi1 25041 |
| This theorem is referenced by: pi1bas 25071 pi1addf 25080 pi1addval 25081 pi1grplem 25082 |
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