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Theorem imasbas 12890
Description: The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)
Hypotheses
Ref Expression
imasbas.u |- ( ph -> U = ( F "s R ) )
imasbas.v |- ( ph -> V = ( Base ` R ) )
imasbas.f |- ( ph -> F : V -onto-> B )
imasbas.r |- ( ph -> R e. Z )
Assertion
Ref Expression
imasbas |- ( ph -> B = ( Base ` U ) )
Proof of Theorem imasbas
Dummy variables p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . . 4 |- ( ph -> U = ( F "s R ) )
2 imasbas.v . . . 4 |- ( ph -> V = ( Base ` R ) )
3 eqid 2193 . . . 4 |- ( +g ` R ) = ( +g ` R )
4 eqid 2193 . . . 4 |- ( .r ` R ) = ( .r ` R )
5 eqid 2193 . . . 4 |- ( .s ` R ) = ( .s ` R )
6 eqidd 2194 . . . 4 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
7 eqidd 2194 . . . 4 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
8 imasbas.f . . . 4 |- ( ph -> F : V -onto-> B )
9 imasbas.r . . . 4 |- ( ph -> R e. Z )
101, 2, 3, 4, 5, 6, 7, 8, 9imasival 12889 . . 3 |- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
1110fveq1d 5556 . 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } ` ( Base ` ndx ) ) )
12 basendxnn 12674 . . . . . 6 |- ( Base ` ndx ) e. NN
13 basfn 12676 . . . . . . . . 9 |- Base Fn _V
149elexd 2773 . . . . . . . . 9 |- ( ph -> R e. _V )
15 funfvex 5571 . . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
1615funfni 5354 . . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
1713, 14, 16sylancr 414 . . . . . . . 8 |- ( ph -> ( Base ` R ) e. _V )
182, 17eqeltrd 2270 . . . . . . 7 |- ( ph -> V e. _V )
19 focdmex 6167 . . . . . . 7 |- ( V e. _V -> ( F : V -onto-> B -> B e. _V ) )
2018, 8, 19sylc 62 . . . . . 6 |- ( ph -> B e. _V )
21 opexg 4257 . . . . . 6 |- ( ( ( Base ` ndx ) e. NN /\ B e. _V ) -> <. ( Base ` ndx ) , B >. e. _V )
2212, 20, 21sylancr 414 . . . . 5 |- ( ph -> <. ( Base ` ndx ) , B >. e. _V )
23 plusgndxnn 12729 . . . . . 6 |- ( +g ` ndx ) e. NN
24 fof 5476 . . . . . . . . . . . . . . . 16 |- ( F : V -onto-> B -> F : V --> B )
258, 24syl 14 . . . . . . . . . . . . . . 15 |- ( ph -> F : V --> B )
2625, 18fexd 5788 . . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
27 vex 2763 . . . . . . . . . . . . . 14 |- p e. _V
28 fvexg 5573 . . . . . . . . . . . . . 14 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
2926, 27, 28sylancl 413 . . . . . . . . . . . . 13 |- ( ph -> ( F ` p ) e. _V )
30 vex 2763 . . . . . . . . . . . . . 14 |- q e. _V
31 fvexg 5573 . . . . . . . . . . . . . 14 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
3226, 30, 31sylancl 413 . . . . . . . . . . . . 13 |- ( ph -> ( F ` q ) e. _V )
33 opexg 4257 . . . . . . . . . . . . 13 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
3429, 32, 33syl2anc 411 . . . . . . . . . . . 12 |- ( ph -> <. ( F ` p ) , ( F ` q ) >. e. _V )
3527a1i 9 . . . . . . . . . . . . . 14 |- ( ph -> p e. _V )
36 plusgslid 12730 . . . . . . . . . . . . . . . 16 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
3736slotex 12645 . . . . . . . . . . . . . . 15 |- ( R e. Z -> ( +g ` R ) e. _V )
389, 37syl 14 . . . . . . . . . . . . . 14 |- ( ph -> ( +g ` R ) e. _V )
3930a1i 9 . . . . . . . . . . . . . 14 |- ( ph -> q e. _V )
40 ovexg 5952 . . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
4135, 38, 39, 40syl3anc 1249 . . . . . . . . . . . . 13 |- ( ph -> ( p ( +g ` R ) q ) e. _V )
42 fvexg 5573 . . . . . . . . . . . . 13 |- ( ( F e. _V /\ ( p ( +g ` R ) q ) e. _V ) -> ( F ` ( p ( +g ` R ) q ) ) e. _V )
4326, 41, 42syl2anc 411 . . . . . . . . . . . 12 |- ( ph -> ( F ` ( p ( +g ` R ) q ) ) e. _V )
44 opexg 4257 . . . . . . . . . . . 12 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p ( +g ` R ) q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
4534, 43, 44syl2anc 411 . . . . . . . . . . 11 |- ( ph -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
46 snexg 4213 . . . . . . . . . . 11 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
4745, 46syl 14 . . . . . . . . . 10 |- ( ph -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
4847ralrimivw 2568 . . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
49 iunexg 6171 . . . . . . . . 9 |- ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
5018, 48, 49syl2anc 411 . . . . . . . 8 |- ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
5150ralrimivw 2568 . . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
52 iunexg 6171 . . . . . . 7 |- ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
5318, 51, 52syl2anc 411 . . . . . 6 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
54 opexg 4257 . . . . . 6 |- ( ( ( +g ` ndx ) e. NN /\ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V )
5523, 53, 54sylancr 414 . . . . 5 |- ( ph -> <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V )
56 mulrslid 12749 . . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
5756simpri 113 . . . . . 6 |- ( .r ` ndx ) e. NN
5856slotex 12645 . . . . . . . . . . . . . . 15 |- ( R e. Z -> ( .r ` R ) e. _V )
599, 58syl 14 . . . . . . . . . . . . . 14 |- ( ph -> ( .r ` R ) e. _V )
60 ovexg 5952 . . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
6135, 59, 39, 60syl3anc 1249 . . . . . . . . . . . . 13 |- ( ph -> ( p ( .r ` R ) q ) e. _V )
62 fvexg 5573 . . . . . . . . . . . . 13 |- ( ( F e. _V /\ ( p ( .r ` R ) q ) e. _V ) -> ( F ` ( p ( .r ` R ) q ) ) e. _V )
6326, 61, 62syl2anc 411 . . . . . . . . . . . 12 |- ( ph -> ( F ` ( p ( .r ` R ) q ) ) e. _V )
64 opexg 4257 . . . . . . . . . . . 12 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p ( .r ` R ) q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
6534, 63, 64syl2anc 411 . . . . . . . . . . 11 |- ( ph -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
66 snexg 4213 . . . . . . . . . . 11 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
6765, 66syl 14 . . . . . . . . . 10 |- ( ph -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
6867ralrimivw 2568 . . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
69 iunexg 6171 . . . . . . . . 9 |- ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
7018, 68, 69syl2anc 411 . . . . . . . 8 |- ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
7170ralrimivw 2568 . . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
72 iunexg 6171 . . . . . . 7 |- ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
7318, 71, 72syl2anc 411 . . . . . 6 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
74 opexg 4257 . . . . . 6 |- ( ( ( .r ` ndx ) e. NN /\ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V )
7557, 73, 74sylancr 414 . . . . 5 |- ( ph -> <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V )
76 tpexg 4475 . . . . 5 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V /\ <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
7722, 55, 75, 76syl3anc 1249 . . . 4 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
7810, 77eqeltrd 2270 . . 3 |- ( ph -> U e. _V )
79 baseid 12672 . . 3 |- Base = Slot ( Base ` ndx )
8078, 79, 12strndxid 12646 . 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( Base ` U ) )
8112a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) e. NN )
82 basendxnplusgndx 12742 . . . 4 |- ( Base ` ndx ) =/= ( +g ` ndx )
8382a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) =/= ( +g ` ndx ) )
84 basendxnmulrndx 12751 . . . 4 |- ( Base ` ndx ) =/= ( .r ` ndx )
8584a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
86 fvtp1g 5766 . . 3 |- ( ( ( ( Base ` ndx ) e. NN /\ B e. _V ) /\ ( ( Base ` ndx ) =/= ( +g ` ndx ) /\ ( Base ` ndx ) =/= ( .r ` ndx ) ) ) -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } ` ( Base ` ndx ) ) = B )
8781, 20, 83, 85, 86syl22anc 1250 . 2 |- ( ph -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } ` ( Base ` ndx ) ) = B )
8811, 80, 873eqtr3rd 2235 1 |- ( ph -> B = ( Base ` U ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   =/= wne 2364   A.wral 2472   _Vcvv 2760   {csn 3618   {ctp 3620   <.cop 3621   U_ciun 3912   Fn wfn 5249   -->wf 5250   -onto->wfo 5252   ` cfv 5254  (class class class)co 5918   NNcn 8982   ndxcnx 12615  Slot cslot 12617   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   .scvsca 12699   "s cimas 12882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mulr 12709  df-iimas 12885
This theorem is referenced by:  qusbas  12910  imasgrp2  13180  imasabl  13406  imasrng  13452  imasring  13560
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