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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmlift2lem3 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift2 35301. (Contributed by Mario Carneiro, 7-May-2015.) |
Ref | Expression |
---|---|
cvmlift2.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmlift2.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmlift2.g | ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) |
cvmlift2.p | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
cvmlift2.i | ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) |
cvmlift2.h | ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) |
cvmlift2lem3.1 | ⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋))) |
Ref | Expression |
---|---|
cvmlift2lem3 | ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmlift2.b | . 2 ⊢ 𝐵 = ∪ 𝐶 | |
2 | cvmlift2lem3.1 | . 2 ⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋))) | |
3 | cvmlift2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
4 | 3 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
5 | iitopon 24919 | . . . 4 ⊢ II ∈ (TopOn‘(0[,]1)) | |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → II ∈ (TopOn‘(0[,]1))) |
7 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → 𝑋 ∈ (0[,]1)) | |
8 | 6, 6, 7 | cnmptc 23686 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ 𝑋) ∈ (II Cn II)) |
9 | 6 | cnmptid 23685 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ 𝑧) ∈ (II Cn II)) |
10 | cvmlift2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → 𝐺 ∈ ((II ×t II) Cn 𝐽)) |
12 | 6, 8, 9, 11 | cnmpt12f 23690 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∈ (II Cn 𝐽)) |
13 | cvmlift2.p | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
14 | cvmlift2.i | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) | |
15 | cvmlift2.h | . . . . . 6 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) | |
16 | 1, 3, 10, 13, 14, 15 | cvmlift2lem2 35289 | . . . . 5 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃)) |
17 | 16 | simp1d 1141 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐶)) |
18 | iiuni 24921 | . . . . 5 ⊢ (0[,]1) = ∪ II | |
19 | 18, 1 | cnf 23270 | . . . 4 ⊢ (𝐻 ∈ (II Cn 𝐶) → 𝐻:(0[,]1)⟶𝐵) |
20 | 17, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻:(0[,]1)⟶𝐵) |
21 | 20 | ffvelcdmda 7104 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐻‘𝑋) ∈ 𝐵) |
22 | 0elunit 13506 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
23 | oveq2 7439 | . . . . 5 ⊢ (𝑧 = 0 → (𝑋𝐺𝑧) = (𝑋𝐺0)) | |
24 | eqid 2735 | . . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) | |
25 | ovex 7464 | . . . . 5 ⊢ (𝑋𝐺0) ∈ V | |
26 | 23, 24, 25 | fvmpt 7016 | . . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0) = (𝑋𝐺0)) |
27 | 22, 26 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0) = (𝑋𝐺0)) |
28 | 16 | simp2d 1142 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))) |
29 | 28 | fveq1d 6909 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋)) |
30 | oveq1 7438 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧𝐺0) = (𝑋𝐺0)) | |
31 | eqid 2735 | . . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) | |
32 | 30, 31, 25 | fvmpt 7016 | . . . 4 ⊢ (𝑋 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋) = (𝑋𝐺0)) |
33 | 29, 32 | sylan9eq 2795 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝑋𝐺0)) |
34 | fvco3 7008 | . . . 4 ⊢ ((𝐻:(0[,]1)⟶𝐵 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋))) | |
35 | 20, 34 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋))) |
36 | 27, 33, 35 | 3eqtr2rd 2782 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐹‘(𝐻‘𝑋)) = ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0)) |
37 | 1, 2, 4, 12, 21, 36 | cvmliftiota 35286 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 0cc0 11153 1c1 11154 [,]cicc 13387 TopOnctopon 22932 Cn ccn 23248 ×t ctx 23584 IIcii 24915 CovMap ccvm 35240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-cn 23251 df-cnp 23252 df-cmp 23411 df-conn 23436 df-lly 23490 df-nlly 23491 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-ii 24917 df-cncf 24918 df-htpy 25016 df-phtpy 25017 df-phtpc 25038 df-pconn 35206 df-sconn 35207 df-cvm 35241 |
This theorem is referenced by: cvmlift2lem5 35292 cvmlift2lem6 35293 cvmlift2lem7 35294 cvmlift2lem8 35295 |
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