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Mirrors > Home > MPE Home > Th. List > pi1bas3 | Structured version Visualization version GIF version |
Description: The base set of the fundamental group. (Contributed 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) |
pi1bas2.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
pi1bas3.r | ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) |
Ref | Expression |
---|---|
pi1bas3 | ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1val.g | . . . 4 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
2 | pi1val.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | pi1val.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
4 | pi1bas2.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
5 | 1, 2, 3, 4 | pi1bas2 23572 | . . 3 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃ph‘𝐽))) |
6 | eqid 2818 | . . . . . 6 ⊢ (𝐽 Ω1 𝑌) = (𝐽 Ω1 𝑌) | |
7 | eqidd 2819 | . . . . . . 7 ⊢ (𝜑 → (Base‘(𝐽 Ω1 𝑌)) = (Base‘(𝐽 Ω1 𝑌))) | |
8 | 1, 2, 3, 6, 4, 7 | pi1buni 23571 | . . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = (Base‘(𝐽 Ω1 𝑌))) |
9 | 1, 2, 3, 6, 4, 8 | pi1blem 23570 | . . . . 5 ⊢ (𝜑 → ((( ≃ph‘𝐽) “ ∪ 𝐵) ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ (II Cn 𝐽))) |
10 | 9 | simpld 495 | . . . 4 ⊢ (𝜑 → (( ≃ph‘𝐽) “ ∪ 𝐵) ⊆ ∪ 𝐵) |
11 | qsinxp 8362 | . . . 4 ⊢ ((( ≃ph‘𝐽) “ ∪ 𝐵) ⊆ ∪ 𝐵 → (∪ 𝐵 / ( ≃ph‘𝐽)) = (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (∪ 𝐵 / ( ≃ph‘𝐽)) = (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) |
13 | 5, 12 | eqtrd 2853 | . 2 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) |
14 | pi1bas3.r | . . 3 ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) | |
15 | qseq2 8333 | . . 3 ⊢ (𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) → (∪ 𝐵 / 𝑅) = (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) | |
16 | 14, 15 | ax-mp 5 | . 2 ⊢ (∪ 𝐵 / 𝑅) = (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))) |
17 | 13, 16 | syl6eqr 2871 | 1 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∪ cuni 4830 × cxp 5546 “ cima 5551 ‘cfv 6348 (class class class)co 7145 / cqs 8277 Basecbs 16471 TopOnctopon 21446 Cn ccn 21760 IIcii 23410 ≃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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 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-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-qus 16770 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-ii 23412 df-htpy 23501 df-phtpy 23502 df-phtpc 23523 df-om1 23537 df-pi1 23539 |
This theorem is referenced by: pi1addf 23578 |
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