ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imasbas Unicode version
Theorem imasbas 13380
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 2229 . . . 4 |- ( +g ` R ) = ( +g ` R )
4 eqid 2229 . . . 4 |- ( .r ` R ) = ( .r ` R )
5 eqid 2229 . . . 4 |- ( .s ` R ) = ( .s ` R )
6 eqidd 2230 . . . 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 2230 . . . 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 13379 . . 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 5637 . 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 13128 . . . . . 6 |- ( Base ` ndx ) e. NN
13 basfn 13131 . . . . . . . . 9 |- Base Fn _V
149elexd 2814 . . . . . . . . 9 |- ( ph -> R e. _V )
15 funfvex 5652 . . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
1615funfni 5429 . . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
1713, 14, 16sylancr 414 . . . . . . . 8 |- ( ph -> ( Base ` R ) e. _V )
182, 17eqeltrd 2306 . . . . . . 7 |- ( ph -> V e. _V )
19 focdmex 6272 . . . . . . 7 |- ( V e. _V -> ( F : V -onto-> B -> B e. _V ) )
2018, 8, 19sylc 62 . . . . . 6 |- ( ph -> B e. _V )
21 opexg 4318 . . . . . 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 13184 . . . . . 6 |- ( +g ` ndx ) e. NN
24 fof 5556 . . . . . . . . . . . . . . . 16 |- ( F : V -onto-> B -> F : V --> B )
258, 24syl 14 . . . . . . . . . . . . . . 15 |- ( ph -> F : V --> B )
2625, 18fexd 5879 . . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
27 vex 2803 . . . . . . . . . . . . . 14 |- p e. _V
28 fvexg 5654 . . . . . . . . . . . . . 14 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
2926, 27, 28sylancl 413 . . . . . . . . . . . . 13 |- ( ph -> ( F ` p ) e. _V )
30 vex 2803 . . . . . . . . . . . . . 14 |- q e. _V
31 fvexg 5654 . . . . . . . . . . . . . 14 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
3226, 30, 31sylancl 413 . . . . . . . . . . . . 13 |- ( ph -> ( F ` q ) e. _V )
33 opexg 4318 . . . . . . . . . . . . 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 13185 . . . . . . . . . . . . . . . 16 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
3736slotex 13099 . . . . . . . . . . . . . . 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 6047 . . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
4135, 38, 39, 40syl3anc 1271 . . . . . . . . . . . . 13 |- ( ph -> ( p ( +g ` R ) q ) e. _V )
42 fvexg 5654 . . . . . . . . . . . . 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 4318 . . . . . . . . . . . 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 4272 . . . . . . . . . . 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 2604 . . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
49 iunexg 6276 . . . . . . . . 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 2604 . . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
52 iunexg 6276 . . . . . . 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 4318 . . . . . 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 13205 . . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
5756simpri 113 . . . . . 6 |- ( .r ` ndx ) e. NN
5856slotex 13099 . . . . . . . . . . . . . . 15 |- ( R e. Z -> ( .r ` R ) e. _V )
599, 58syl 14 . . . . . . . . . . . . . 14 |- ( ph -> ( .r ` R ) e. _V )
60 ovexg 6047 . . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
6135, 59, 39, 60syl3anc 1271 . . . . . . . . . . . . 13 |- ( ph -> ( p ( .r ` R ) q ) e. _V )
62 fvexg 5654 . . . . . . . . . . . . 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 4318 . . . . . . . . . . . 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 4272 . . . . . . . . . . 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 2604 . . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
69 iunexg 6276 . . . . . . . . 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 2604 . . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
72 iunexg 6276 . . . . . . 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 4318 . . . . . 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 4539 . . . . 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 1271 . . . 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 2306 . . 3 |- ( ph -> U e. _V )
79 baseid 13126 . . 3 |- Base = Slot ( Base ` ndx )
8078, 79, 12strndxid 13100 . 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( Base ` U ) )
8112a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) e. NN )
82 basendxnplusgndx 13198 . . . 4 |- ( Base ` ndx ) =/= ( +g ` ndx )
8382a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) =/= ( +g ` ndx ) )
84 basendxnmulrndx 13207 . . . 4 |- ( Base ` ndx ) =/= ( .r ` ndx )
8584a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
86 fvtp1g 5857 . . 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 1272 . 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 2271 1 |- ( ph -> B = ( Base ` U ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   _Vcvv 2800   {csn 3667   {ctp 3669   <.cop 3670   U_ciun 3968   Fn wfn 5319   -->wf 5320   -onto->wfo 5322   ` cfv 5324  (class class class)co 6013   NNcn 9133   ndxcnx 13069  Slot cslot 13071   Basecbs 13072   +g cplusg 13150   .rcmulr 13151   .scvsca 13154   "s cimas 13372
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-mulr 13164  df-iimas 13375
This theorem is referenced by:  qusbas  13400  imasmnd2  13525  imasgrp2  13687  imasabl  13913  imasrng  13959  imasring  14067
  Copyright terms: Public domain W3C validator