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Theorem imasbas 13389
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 2231 . . . 4 |- ( +g ` R ) = ( +g ` R )
4 eqid 2231 . . . 4 |- ( .r ` R ) = ( .r ` R )
5 eqid 2231 . . . 4 |- ( .s ` R ) = ( .s ` R )
6 eqidd 2232 . . . 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 2232 . . . 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 13388 . . 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 5641 . 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 13137 . . . . . 6 |- ( Base ` ndx ) e. NN
13 basfn 13140 . . . . . . . . 9 |- Base Fn _V
149elexd 2816 . . . . . . . . 9 |- ( ph -> R e. _V )
15 funfvex 5656 . . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
1615funfni 5432 . . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
1713, 14, 16sylancr 414 . . . . . . . 8 |- ( ph -> ( Base ` R ) e. _V )
182, 17eqeltrd 2308 . . . . . . 7 |- ( ph -> V e. _V )
19 focdmex 6276 . . . . . . 7 |- ( V e. _V -> ( F : V -onto-> B -> B e. _V ) )
2018, 8, 19sylc 62 . . . . . 6 |- ( ph -> B e. _V )
21 opexg 4320 . . . . . 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 13193 . . . . . 6 |- ( +g ` ndx ) e. NN
24 fof 5559 . . . . . . . . . . . . . . . 16 |- ( F : V -onto-> B -> F : V --> B )
258, 24syl 14 . . . . . . . . . . . . . . 15 |- ( ph -> F : V --> B )
2625, 18fexd 5883 . . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
27 vex 2805 . . . . . . . . . . . . . 14 |- p e. _V
28 fvexg 5658 . . . . . . . . . . . . . 14 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
2926, 27, 28sylancl 413 . . . . . . . . . . . . 13 |- ( ph -> ( F ` p ) e. _V )
30 vex 2805 . . . . . . . . . . . . . 14 |- q e. _V
31 fvexg 5658 . . . . . . . . . . . . . 14 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
3226, 30, 31sylancl 413 . . . . . . . . . . . . 13 |- ( ph -> ( F ` q ) e. _V )
33 opexg 4320 . . . . . . . . . . . . 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 13194 . . . . . . . . . . . . . . . 16 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
3736slotex 13108 . . . . . . . . . . . . . . 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 6051 . . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
4135, 38, 39, 40syl3anc 1273 . . . . . . . . . . . . 13 |- ( ph -> ( p ( +g ` R ) q ) e. _V )
42 fvexg 5658 . . . . . . . . . . . . 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 4320 . . . . . . . . . . . 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 4274 . . . . . . . . . . 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 2606 . . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
49 iunexg 6280 . . . . . . . . 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 2606 . . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
52 iunexg 6280 . . . . . . 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 4320 . . . . . 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 13214 . . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
5756simpri 113 . . . . . 6 |- ( .r ` ndx ) e. NN
5856slotex 13108 . . . . . . . . . . . . . . 15 |- ( R e. Z -> ( .r ` R ) e. _V )
599, 58syl 14 . . . . . . . . . . . . . 14 |- ( ph -> ( .r ` R ) e. _V )
60 ovexg 6051 . . . . . . . . . . . . . 14 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
6135, 59, 39, 60syl3anc 1273 . . . . . . . . . . . . 13 |- ( ph -> ( p ( .r ` R ) q ) e. _V )
62 fvexg 5658 . . . . . . . . . . . . 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 4320 . . . . . . . . . . . 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 4274 . . . . . . . . . . 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 2606 . . . . . . . . 9 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
69 iunexg 6280 . . . . . . . . 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 2606 . . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
72 iunexg 6280 . . . . . . 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 4320 . . . . . 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 4541 . . . . 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 1273 . . . 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 2308 . . 3 |- ( ph -> U e. _V )
79 baseid 13135 . . 3 |- Base = Slot ( Base ` ndx )
8078, 79, 12strndxid 13109 . 2 |- ( ph -> ( U ` ( Base ` ndx ) ) = ( Base ` U ) )
8112a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) e. NN )
82 basendxnplusgndx 13207 . . . 4 |- ( Base ` ndx ) =/= ( +g ` ndx )
8382a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) =/= ( +g ` ndx ) )
84 basendxnmulrndx 13216 . . . 4 |- ( Base ` ndx ) =/= ( .r ` ndx )
8584a1i 9 . . 3 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
86 fvtp1g 5861 . . 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 1274 . 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 2273 1 |- ( ph -> B = ( Base ` U ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   _Vcvv 2802   {csn 3669   {ctp 3671   <.cop 3672   U_ciun 3970   Fn wfn 5321   -->wf 5322   -onto->wfo 5324   ` cfv 5326  (class class class)co 6017   NNcn 9142   ndxcnx 13078  Slot cslot 13080   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   .scvsca 13163   "s cimas 13381
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mulr 13173  df-iimas 13384
This theorem is referenced by:  qusbas  13409  imasmnd2  13534  imasgrp2  13696  imasabl  13922  imasrng  13968  imasring  14076
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