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Mirrors > Home > MPE Home > Th. List > pwsbas | Structured version Visualization version GIF version |
Description: Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsbas.y | β’ π = (π βs πΌ) |
pwsbas.f | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
pwsbas | β’ ((π β π β§ πΌ β π) β (π΅ βm πΌ) = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsbas.y | . . . 4 β’ π = (π βs πΌ) | |
2 | eqid 2728 | . . . 4 β’ (Scalarβπ ) = (Scalarβπ ) | |
3 | 1, 2 | pwsval 17468 | . . 3 β’ ((π β π β§ πΌ β π) β π = ((Scalarβπ )Xs(πΌ Γ {π }))) |
4 | 3 | fveq2d 6901 | . 2 β’ ((π β π β§ πΌ β π) β (Baseβπ) = (Baseβ((Scalarβπ )Xs(πΌ Γ {π })))) |
5 | eqid 2728 | . . . 4 β’ ((Scalarβπ )Xs(πΌ Γ {π })) = ((Scalarβπ )Xs(πΌ Γ {π })) | |
6 | fvexd 6912 | . . . 4 β’ ((π β π β§ πΌ β π) β (Scalarβπ ) β V) | |
7 | simpr 484 | . . . . 5 β’ ((π β π β§ πΌ β π) β πΌ β π) | |
8 | snex 5433 | . . . . 5 β’ {π } β V | |
9 | xpexg 7752 | . . . . 5 β’ ((πΌ β π β§ {π } β V) β (πΌ Γ {π }) β V) | |
10 | 7, 8, 9 | sylancl 585 | . . . 4 β’ ((π β π β§ πΌ β π) β (πΌ Γ {π }) β V) |
11 | eqid 2728 | . . . 4 β’ (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) = (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) | |
12 | snnzg 4779 | . . . . . 6 β’ (π β π β {π } β β ) | |
13 | 12 | adantr 480 | . . . . 5 β’ ((π β π β§ πΌ β π) β {π } β β ) |
14 | dmxp 5931 | . . . . 5 β’ ({π } β β β dom (πΌ Γ {π }) = πΌ) | |
15 | 13, 14 | syl 17 | . . . 4 β’ ((π β π β§ πΌ β π) β dom (πΌ Γ {π }) = πΌ) |
16 | 5, 6, 10, 11, 15 | prdsbas 17439 | . . 3 β’ ((π β π β§ πΌ β π) β (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) = Xπ₯ β πΌ (Baseβ((πΌ Γ {π })βπ₯))) |
17 | fvconst2g 7214 | . . . . . . 7 β’ ((π β π β§ π₯ β πΌ) β ((πΌ Γ {π })βπ₯) = π ) | |
18 | 17 | fveq2d 6901 | . . . . . 6 β’ ((π β π β§ π₯ β πΌ) β (Baseβ((πΌ Γ {π })βπ₯)) = (Baseβπ )) |
19 | 18 | ralrimiva 3143 | . . . . 5 β’ (π β π β βπ₯ β πΌ (Baseβ((πΌ Γ {π })βπ₯)) = (Baseβπ )) |
20 | 19 | adantr 480 | . . . 4 β’ ((π β π β§ πΌ β π) β βπ₯ β πΌ (Baseβ((πΌ Γ {π })βπ₯)) = (Baseβπ )) |
21 | ixpeq2 8930 | . . . 4 β’ (βπ₯ β πΌ (Baseβ((πΌ Γ {π })βπ₯)) = (Baseβπ ) β Xπ₯ β πΌ (Baseβ((πΌ Γ {π })βπ₯)) = Xπ₯ β πΌ (Baseβπ )) | |
22 | 20, 21 | syl 17 | . . 3 β’ ((π β π β§ πΌ β π) β Xπ₯ β πΌ (Baseβ((πΌ Γ {π })βπ₯)) = Xπ₯ β πΌ (Baseβπ )) |
23 | 16, 22 | eqtrd 2768 | . 2 β’ ((π β π β§ πΌ β π) β (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) = Xπ₯ β πΌ (Baseβπ )) |
24 | fvex 6910 | . . . 4 β’ (Baseβπ ) β V | |
25 | ixpconstg 8925 | . . . 4 β’ ((πΌ β π β§ (Baseβπ ) β V) β Xπ₯ β πΌ (Baseβπ ) = ((Baseβπ ) βm πΌ)) | |
26 | 7, 24, 25 | sylancl 585 | . . 3 β’ ((π β π β§ πΌ β π) β Xπ₯ β πΌ (Baseβπ ) = ((Baseβπ ) βm πΌ)) |
27 | pwsbas.f | . . . 4 β’ π΅ = (Baseβπ ) | |
28 | 27 | oveq1i 7430 | . . 3 β’ (π΅ βm πΌ) = ((Baseβπ ) βm πΌ) |
29 | 26, 28 | eqtr4di 2786 | . 2 β’ ((π β π β§ πΌ β π) β Xπ₯ β πΌ (Baseβπ ) = (π΅ βm πΌ)) |
30 | 4, 23, 29 | 3eqtrrd 2773 | 1 β’ ((π β π β§ πΌ β π) β (π΅ βm πΌ) = (Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 Vcvv 3471 β c0 4323 {csn 4629 Γ cxp 5676 dom cdm 5678 βcfv 6548 (class class class)co 7420 βm cmap 8845 Xcixp 8916 Basecbs 17180 Scalarcsca 17236 Xscprds 17427 βs cpws 17428 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-prds 17429 df-pws 17431 |
This theorem is referenced by: pwselbasb 17470 pwssnf1o 17480 pwsdiagmhm 18783 pwsco1rhm 20441 pwsco2rhm 20442 frlmbas 21689 frlmsubgval 21699 psrgrp 21899 evls1val 22239 evls1rhmlem 22240 evl1val 22248 repwsmet 37307 rrnequiv 37308 aks6d1c2lem4 41598 aks6d1c6lem2 41643 psrmnd 41775 mhphf2 41831 pwslnmlem0 42515 |
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