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Mirrors > Home > MPE Home > Th. List > xpsrnbas | Structured version Visualization version GIF version |
Description: The indexed structure product that appears in xpsval 17512 has the same base as the target of the function πΉ. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
Ref | Expression |
---|---|
xpsval.t | β’ π = (π Γs π) |
xpsval.x | β’ π = (Baseβπ ) |
xpsval.y | β’ π = (Baseβπ) |
xpsval.1 | β’ (π β π β π) |
xpsval.2 | β’ (π β π β π) |
xpsval.f | β’ πΉ = (π₯ β π, π¦ β π β¦ {β¨β , π₯β©, β¨1o, π¦β©}) |
xpsval.k | β’ πΊ = (Scalarβπ ) |
xpsval.u | β’ π = (πΊXs{β¨β , π β©, β¨1o, πβ©}) |
Ref | Expression |
---|---|
xpsrnbas | β’ (π β ran πΉ = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsval.u | . . 3 β’ π = (πΊXs{β¨β , π β©, β¨1o, πβ©}) | |
2 | eqid 2724 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
3 | xpsval.k | . . . . 5 β’ πΊ = (Scalarβπ ) | |
4 | 3 | fvexi 6895 | . . . 4 β’ πΊ β V |
5 | 4 | a1i 11 | . . 3 β’ (π β πΊ β V) |
6 | 2on 8475 | . . . 4 β’ 2o β On | |
7 | 6 | a1i 11 | . . 3 β’ (π β 2o β On) |
8 | xpsval.1 | . . . 4 β’ (π β π β π) | |
9 | xpsval.2 | . . . 4 β’ (π β π β π) | |
10 | fnpr2o 17499 | . . . 4 β’ ((π β π β§ π β π) β {β¨β , π β©, β¨1o, πβ©} Fn 2o) | |
11 | 8, 9, 10 | syl2anc 583 | . . 3 β’ (π β {β¨β , π β©, β¨1o, πβ©} Fn 2o) |
12 | 1, 2, 5, 7, 11 | prdsbas2 17411 | . 2 β’ (π β (Baseβπ) = Xπ β 2o (Baseβ({β¨β , π β©, β¨1o, πβ©}βπ))) |
13 | fvprif 17503 | . . . . . . . . 9 β’ ((π β π β§ π β π β§ π β 2o) β ({β¨β , π β©, β¨1o, πβ©}βπ) = if(π = β , π , π)) | |
14 | 13 | 3expia 1118 | . . . . . . . 8 β’ ((π β π β§ π β π) β (π β 2o β ({β¨β , π β©, β¨1o, πβ©}βπ) = if(π = β , π , π))) |
15 | 8, 9, 14 | syl2anc 583 | . . . . . . 7 β’ (π β (π β 2o β ({β¨β , π β©, β¨1o, πβ©}βπ) = if(π = β , π , π))) |
16 | 15 | imp 406 | . . . . . 6 β’ ((π β§ π β 2o) β ({β¨β , π β©, β¨1o, πβ©}βπ) = if(π = β , π , π)) |
17 | 16 | fveq2d 6885 | . . . . 5 β’ ((π β§ π β 2o) β (Baseβ({β¨β , π β©, β¨1o, πβ©}βπ)) = (Baseβif(π = β , π , π))) |
18 | xpsval.x | . . . . . . 7 β’ π = (Baseβπ ) | |
19 | xpsval.y | . . . . . . 7 β’ π = (Baseβπ) | |
20 | ifeq12 4538 | . . . . . . 7 β’ ((π = (Baseβπ ) β§ π = (Baseβπ)) β if(π = β , π, π) = if(π = β , (Baseβπ ), (Baseβπ))) | |
21 | 18, 19, 20 | mp2an 689 | . . . . . 6 β’ if(π = β , π, π) = if(π = β , (Baseβπ ), (Baseβπ)) |
22 | fvif 6897 | . . . . . 6 β’ (Baseβif(π = β , π , π)) = if(π = β , (Baseβπ ), (Baseβπ)) | |
23 | 21, 22 | eqtr4i 2755 | . . . . 5 β’ if(π = β , π, π) = (Baseβif(π = β , π , π)) |
24 | 17, 23 | eqtr4di 2782 | . . . 4 β’ ((π β§ π β 2o) β (Baseβ({β¨β , π β©, β¨1o, πβ©}βπ)) = if(π = β , π, π)) |
25 | 24 | ixpeq2dva 8901 | . . 3 β’ (π β Xπ β 2o (Baseβ({β¨β , π β©, β¨1o, πβ©}βπ)) = Xπ β 2o if(π = β , π, π)) |
26 | xpsval.f | . . . 4 β’ πΉ = (π₯ β π, π¦ β π β¦ {β¨β , π₯β©, β¨1o, π¦β©}) | |
27 | 26 | xpsfrn 17510 | . . 3 β’ ran πΉ = Xπ β 2o if(π = β , π, π) |
28 | 25, 27 | eqtr4di 2782 | . 2 β’ (π β Xπ β 2o (Baseβ({β¨β , π β©, β¨1o, πβ©}βπ)) = ran πΉ) |
29 | 12, 28 | eqtr2d 2765 | 1 β’ (π β ran πΉ = (Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3466 β c0 4314 ifcif 4520 {cpr 4622 β¨cop 4626 ran crn 5667 Oncon0 6354 Fn wfn 6528 βcfv 6533 (class class class)co 7401 β cmpo 7403 1oc1o 8454 2oc2o 8455 Xcixp 8886 Basecbs 17140 Scalarcsca 17196 Xscprds 17387 Γs cxps 17448 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-prds 17389 |
This theorem is referenced by: xpsbas 17514 xpsaddlem 17515 xpsadd 17516 xpsmul 17517 xpssca 17518 xpsvsca 17519 xpsless 17520 xpsle 17521 xpsmnd 18694 xpsgrp 18974 xpsrngd 20069 xpsringd 20216 xpstps 23624 xpstopnlem2 23625 xpsdsfn 24193 xpsxmetlem 24195 xpsxmet 24196 xpsdsval 24197 xpsmet 24198 xpsxms 24353 xpsms 24354 |
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