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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmlift3lem1 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift3 34861. (Contributed by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
cvmlift3.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmlift3.y | ⊢ 𝑌 = ∪ 𝐾 |
cvmlift3.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmlift3.k | ⊢ (𝜑 → 𝐾 ∈ SConn) |
cvmlift3.l | ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn) |
cvmlift3.o | ⊢ (𝜑 → 𝑂 ∈ 𝑌) |
cvmlift3.g | ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) |
cvmlift3.p | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
cvmlift3.e | ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂)) |
cvmlift3lem1.1 | ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾)) |
cvmlift3lem1.2 | ⊢ (𝜑 → (𝑀‘0) = 𝑂) |
cvmlift3lem1.3 | ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾)) |
cvmlift3lem1.4 | ⊢ (𝜑 → (𝑁‘0) = 𝑂) |
cvmlift3lem1.5 | ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1)) |
Ref | Expression |
---|---|
cvmlift3lem1 | ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmlift3.b | . . . 4 ⊢ 𝐵 = ∪ 𝐶 | |
2 | eqid 2727 | . . . 4 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃)) | |
3 | eqid 2727 | . . . 4 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃)) | |
4 | cvmlift3.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
5 | cvmlift3.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
6 | cvmlift3.e | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂)) | |
7 | cvmlift3lem1.2 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘0) = 𝑂) | |
8 | 7 | fveq2d 6895 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑀‘0)) = (𝐺‘𝑂)) |
9 | 6, 8 | eqtr4d 2770 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(𝑀‘0))) |
10 | cvmlift3lem1.1 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾)) | |
11 | iiuni 24775 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
12 | cvmlift3.y | . . . . . . . 8 ⊢ 𝑌 = ∪ 𝐾 | |
13 | 11, 12 | cnf 23124 | . . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐾) → 𝑀:(0[,]1)⟶𝑌) |
14 | 10, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝑌) |
15 | 0elunit 13464 | . . . . . 6 ⊢ 0 ∈ (0[,]1) | |
16 | fvco3 6991 | . . . . . 6 ⊢ ((𝑀:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0))) | |
17 | 14, 15, 16 | sylancl 585 | . . . . 5 ⊢ (𝜑 → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0))) |
18 | 9, 17 | eqtr4d 2770 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ 𝑀)‘0)) |
19 | cvmlift3.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ SConn) | |
20 | cvmlift3lem1.3 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾)) | |
21 | cvmlift3lem1.4 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘0) = 𝑂) | |
22 | 7, 21 | eqtr4d 2770 | . . . . . 6 ⊢ (𝜑 → (𝑀‘0) = (𝑁‘0)) |
23 | cvmlift3lem1.5 | . . . . . 6 ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1)) | |
24 | 19, 10, 20, 22, 23 | sconnpht2 34771 | . . . . 5 ⊢ (𝜑 → 𝑀( ≃ph‘𝐾)𝑁) |
25 | cvmlift3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) | |
26 | 24, 25 | phtpcco2 24900 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝑀)( ≃ph‘𝐽)(𝐺 ∘ 𝑁)) |
27 | 1, 2, 3, 4, 5, 18, 26 | cvmliftpht 34851 | . . 3 ⊢ (𝜑 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))( ≃ph‘𝐶)(℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))) |
28 | phtpc01 24896 | . . 3 ⊢ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))( ≃ph‘𝐶)(℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃)) → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘0) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘0) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))) | |
29 | 27, 28 | syl 17 | . 2 ⊢ (𝜑 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘0) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘0) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))) |
30 | 29 | simprd 495 | 1 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∪ cuni 4903 class class class wbr 5142 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 0cc0 11124 1c1 11125 [,]cicc 13345 Cn ccn 23102 𝑛-Locally cnlly 23343 IIcii 24769 ≃phcphtpc 24869 PConncpconn 34752 SConncsconn 34753 CovMap ccvm 34788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-ec 8718 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-sum 15651 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-mulg 19008 df-cntz 19252 df-cmn 19721 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-cn 23105 df-cnp 23106 df-cmp 23265 df-conn 23290 df-lly 23344 df-nlly 23345 df-tx 23440 df-hmeo 23633 df-xms 24200 df-ms 24201 df-tms 24202 df-ii 24771 df-cncf 24772 df-htpy 24870 df-phtpy 24871 df-phtpc 24892 df-pco 24906 df-pconn 34754 df-sconn 34755 df-cvm 34789 |
This theorem is referenced by: cvmlift3lem2 34853 |
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