Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pi1buni | Structured version Visualization version GIF version |
Description: Another way to write the loop space base in terms of the base 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) |
pi1val.o | ⊢ 𝑂 = (𝐽 Ω1 𝑌) |
pi1bas.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
pi1bas.k | ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) |
Ref | Expression |
---|---|
pi1buni | ⊢ (𝜑 → ∪ 𝐵 = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1val.g | . . . . 5 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
2 | pi1val.1 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | pi1val.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
4 | pi1val.o | . . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
5 | pi1bas.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
6 | pi1bas.k | . . . . 5 ⊢ (𝜑 → 𝐾 = (Base‘𝑂)) | |
7 | 1, 2, 3, 4, 5, 6 | pi1bas 24189 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝐾 / ( ≃ph‘𝐽))) |
8 | 1, 2, 3, 4, 5, 6 | pi1blem 24190 | . . . . . 6 ⊢ (𝜑 → ((( ≃ph‘𝐽) “ 𝐾) ⊆ 𝐾 ∧ 𝐾 ⊆ (II Cn 𝐽))) |
9 | 8 | simpld 495 | . . . . 5 ⊢ (𝜑 → (( ≃ph‘𝐽) “ 𝐾) ⊆ 𝐾) |
10 | qsinxp 8570 | . . . . 5 ⊢ ((( ≃ph‘𝐽) “ 𝐾) ⊆ 𝐾 → (𝐾 / ( ≃ph‘𝐽)) = (𝐾 / (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)))) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐾 / ( ≃ph‘𝐽)) = (𝐾 / (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)))) |
12 | 7, 11 | eqtrd 2778 | . . 3 ⊢ (𝜑 → 𝐵 = (𝐾 / (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)))) |
13 | 12 | unieqd 4854 | . 2 ⊢ (𝜑 → ∪ 𝐵 = ∪ (𝐾 / (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)))) |
14 | phtpcer 24146 | . . . . 5 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) |
16 | 8 | simprd 496 | . . . 4 ⊢ (𝜑 → 𝐾 ⊆ (II Cn 𝐽)) |
17 | 15, 16 | erinxp 8568 | . . 3 ⊢ (𝜑 → (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)) Er 𝐾) |
18 | fvex 6780 | . . . . 5 ⊢ ( ≃ph‘𝐽) ∈ V | |
19 | 18 | inex1 5240 | . . . 4 ⊢ (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)) ∈ V |
20 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾)) ∈ V) |
21 | 17, 20 | uniqs2 8556 | . 2 ⊢ (𝜑 → ∪ (𝐾 / (( ≃ph‘𝐽) ∩ (𝐾 × 𝐾))) = 𝐾) |
22 | 13, 21 | eqtrd 2778 | 1 ⊢ (𝜑 → ∪ 𝐵 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3430 ∩ cin 3886 ⊆ wss 3887 ∪ cuni 4840 × cxp 5583 “ cima 5588 ‘cfv 6427 (class class class)co 7268 Er wer 8483 / cqs 8485 Basecbs 16900 TopOnctopon 22047 Cn ccn 22363 IIcii 24026 ≃phcphtpc 24120 Ω1 comi 24152 π1 cpi1 24154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 ax-mulf 10939 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-er 8486 df-ec 8488 df-qs 8492 df-map 8605 df-ixp 8674 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ioo 13071 df-icc 13074 df-fz 13228 df-fzo 13371 df-seq 13710 df-exp 13771 df-hash 14033 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-starv 16965 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-unif 16973 df-hom 16974 df-cco 16975 df-rest 17121 df-topn 17122 df-0g 17140 df-gsum 17141 df-topgen 17142 df-pt 17143 df-prds 17146 df-xrs 17201 df-qtop 17206 df-imas 17207 df-qus 17208 df-xps 17209 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-mulg 18689 df-cntz 18911 df-cmn 19376 df-psmet 20577 df-xmet 20578 df-met 20579 df-bl 20580 df-mopn 20581 df-cnfld 20586 df-top 22031 df-topon 22048 df-topsp 22070 df-bases 22084 df-cld 22158 df-cn 22366 df-cnp 22367 df-tx 22701 df-hmeo 22894 df-xms 23461 df-ms 23462 df-tms 23463 df-ii 24028 df-htpy 24121 df-phtpy 24122 df-phtpc 24143 df-om1 24157 df-pi1 24159 |
This theorem is referenced by: pi1bas2 24192 pi1eluni 24193 pi1bas3 24194 pi1cpbl 24195 pi1addf 24198 pi1addval 24199 pi1grplem 24200 |
Copyright terms: Public domain | W3C validator |