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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmlift3lem1 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift3 34991. (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 2725 | . . . 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 2725 | . . . 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 6894 | . . . . . 6 ⊢ (𝜑 → (𝐺‘(𝑀‘0)) = (𝐺‘𝑂)) |
9 | 6, 8 | eqtr4d 2768 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘(𝑀‘0))) |
10 | cvmlift3lem1.1 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾)) | |
11 | iiuni 24814 | . . . . . . . 8 ⊢ (0[,]1) = ∪ II | |
12 | cvmlift3.y | . . . . . . . 8 ⊢ 𝑌 = ∪ 𝐾 | |
13 | 11, 12 | cnf 23163 | . . . . . . 7 ⊢ (𝑀 ∈ (II Cn 𝐾) → 𝑀:(0[,]1)⟶𝑌) |
14 | 10, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀:(0[,]1)⟶𝑌) |
15 | 0elunit 13473 | . . . . . 6 ⊢ 0 ∈ (0[,]1) | |
16 | fvco3 6990 | . . . . . 6 ⊢ ((𝑀:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0))) | |
17 | 14, 15, 16 | sylancl 584 | . . . . 5 ⊢ (𝜑 → ((𝐺 ∘ 𝑀)‘0) = (𝐺‘(𝑀‘0))) |
18 | 9, 17 | eqtr4d 2768 | . . . 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 2768 | . . . . . 6 ⊢ (𝜑 → (𝑀‘0) = (𝑁‘0)) |
23 | cvmlift3lem1.5 | . . . . . 6 ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1)) | |
24 | 19, 10, 20, 22, 23 | sconnpht2 34901 | . . . . 5 ⊢ (𝜑 → 𝑀( ≃ph‘𝐾)𝑁) |
25 | cvmlift3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) | |
26 | 24, 25 | phtpcco2 24939 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝑀)( ≃ph‘𝐽)(𝐺 ∘ 𝑁)) |
27 | 1, 2, 3, 4, 5, 18, 26 | cvmliftpht 34981 | . . 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 24935 | . . 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 494 | 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 394 = wceq 1533 ∈ wcel 2098 ∪ cuni 4904 class class class wbr 5144 ∘ ccom 5677 ⟶wf 6539 ‘cfv 6543 ℩crio 7368 (class class class)co 7413 0cc0 11133 1c1 11134 [,]cicc 13354 Cn ccn 23141 𝑛-Locally cnlly 23382 IIcii 24808 ≃phcphtpc 24908 PConncpconn 34882 SConncsconn 34883 CovMap ccvm 34918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-ec 8720 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-cnfld 21279 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-cn 23144 df-cnp 23145 df-cmp 23304 df-conn 23329 df-lly 23383 df-nlly 23384 df-tx 23479 df-hmeo 23672 df-xms 24239 df-ms 24240 df-tms 24241 df-ii 24810 df-cncf 24811 df-htpy 24909 df-phtpy 24910 df-phtpc 24931 df-pco 24945 df-pconn 34884 df-sconn 34885 df-cvm 34919 |
This theorem is referenced by: cvmlift3lem2 34983 |
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