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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 2726 | . . . 4 β’ (Scalarβπ ) = (Scalarβπ ) | |
3 | 1, 2 | pwsval 17439 | . . 3 β’ ((π β π β§ πΌ β π) β π = ((Scalarβπ )Xs(πΌ Γ {π }))) |
4 | 3 | fveq2d 6888 | . 2 β’ ((π β π β§ πΌ β π) β (Baseβπ) = (Baseβ((Scalarβπ )Xs(πΌ Γ {π })))) |
5 | eqid 2726 | . . . 4 β’ ((Scalarβπ )Xs(πΌ Γ {π })) = ((Scalarβπ )Xs(πΌ Γ {π })) | |
6 | fvexd 6899 | . . . 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 5424 | . . . . 5 β’ {π } β V | |
9 | xpexg 7733 | . . . . 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 2726 | . . . 4 β’ (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) = (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) | |
12 | snnzg 4773 | . . . . . 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 5921 | . . . . 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 17410 | . . 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 7198 | . . . . . . 7 β’ ((π β π β§ π₯ β πΌ) β ((πΌ Γ {π })βπ₯) = π ) | |
18 | 17 | fveq2d 6888 | . . . . . 6 β’ ((π β π β§ π₯ β πΌ) β (Baseβ((πΌ Γ {π })βπ₯)) = (Baseβπ )) |
19 | 18 | ralrimiva 3140 | . . . . 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 8904 | . . . 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 2766 | . 2 β’ ((π β π β§ πΌ β π) β (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) = Xπ₯ β πΌ (Baseβπ )) |
24 | fvex 6897 | . . . 4 β’ (Baseβπ ) β V | |
25 | ixpconstg 8899 | . . . 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 7414 | . . 3 β’ (π΅ βm πΌ) = ((Baseβπ ) βm πΌ) |
29 | 26, 28 | eqtr4di 2784 | . 2 β’ ((π β π β§ πΌ β π) β Xπ₯ β πΌ (Baseβπ ) = (π΅ βm πΌ)) |
30 | 4, 23, 29 | 3eqtrrd 2771 | 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 1533 β wcel 2098 β wne 2934 βwral 3055 Vcvv 3468 β c0 4317 {csn 4623 Γ cxp 5667 dom cdm 5669 βcfv 6536 (class class class)co 7404 βm cmap 8819 Xcixp 8890 Basecbs 17151 Scalarcsca 17207 Xscprds 17398 βs cpws 17399 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-prds 17400 df-pws 17402 |
This theorem is referenced by: pwselbasb 17441 pwssnf1o 17451 pwsdiagmhm 18754 pwsco1rhm 20402 pwsco2rhm 20403 frlmbas 21646 frlmsubgval 21656 psrgrp 21855 evls1val 22190 evls1rhmlem 22191 evl1val 22199 repwsmet 37213 rrnequiv 37214 mhphf2 41708 pwslnmlem0 42392 |
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