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Theorem rrxip 21573
 Description: The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
rrxval.r ℝ^
rrxbase.b
Assertion
Ref Expression
rrxip RRfld g
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem rrxip
StepHypRef Expression
1 rrxval.r . . . 4 ℝ^
2 rrxbase.b . . . 4
31, 2rrxprds 21572 . . 3 toCHilRRflds subringAlg RRflds
43fveq2d 5869 . 2 toCHilRRflds subringAlg RRflds
5 eqid 2467 . . . 4 toCHilRRflds subringAlg RRflds toCHilRRflds subringAlg RRflds
6 eqid 2467 . . . 4 RRflds subringAlg RRflds RRflds subringAlg RRflds
75, 6tchip 21419 . . 3 RRflds subringAlg RRflds toCHilRRflds subringAlg RRflds
8 fvex 5875 . . . . . 6
92, 8eqeltri 2551 . . . . 5
10 eqid 2467 . . . . . 6 RRflds subringAlg RRflds RRflds subringAlg RRflds
11 eqid 2467 . . . . . 6 RRflds subringAlg RRfld RRflds subringAlg RRfld
1210, 11ressip 14634 . . . . 5 RRflds subringAlg RRfld RRflds subringAlg RRflds
139, 12ax-mp 5 . . . 4 RRflds subringAlg RRfld RRflds subringAlg RRflds
14 eqid 2467 . . . . . 6 RRflds subringAlg RRfld RRflds subringAlg RRfld
15 refld 18438 . . . . . . 7 RRfld Field
1615a1i 11 . . . . . 6 RRfld Field
17 snex 4688 . . . . . . 7 subringAlg RRfld
18 xpexg 6710 . . . . . . 7 subringAlg RRfld subringAlg RRfld
1917, 18mpan2 671 . . . . . 6 subringAlg RRfld
20 eqid 2467 . . . . . 6 RRflds subringAlg RRfld RRflds subringAlg RRfld
21 fvex 5875 . . . . . . . . 9 subringAlg RRfld
22 snnzb 4092 . . . . . . . . 9 subringAlg RRfld subringAlg RRfld
2321, 22mpbi 208 . . . . . . . 8 subringAlg RRfld
24 dmxp 5220 . . . . . . . 8 subringAlg RRfld subringAlg RRfld
2523, 24ax-mp 5 . . . . . . 7 subringAlg RRfld
2625a1i 11 . . . . . 6 subringAlg RRfld
2714, 16, 19, 20, 26, 11prdsip 14715 . . . . 5 RRflds subringAlg RRfld RRflds subringAlg RRfld RRflds subringAlg RRfld RRfld g subringAlg RRfld
2814, 16, 19, 20, 26prdsbas 14711 . . . . . . 7 RRflds subringAlg RRfld subringAlg RRfld
29 eqidd 2468 . . . . . . . . . . 11 subringAlg RRfld subringAlg RRfld
30 ssid 3523 . . . . . . . . . . . . . 14
31 rebase 18425 . . . . . . . . . . . . . 14 RRfld
3230, 31sseqtri 3536 . . . . . . . . . . . . 13 RRfld
3332rgenw 2825 . . . . . . . . . . . 12 RRfld
3433rspec 2832 . . . . . . . . . . 11 RRfld
3529, 34srabase 17619 . . . . . . . . . 10 RRfld subringAlg RRfld
3631a1i 11 . . . . . . . . . 10 RRfld
37 fvconst2g 6113 . . . . . . . . . . . 12 subringAlg RRfld subringAlg RRfld subringAlg RRfld
3821, 37mpan 670 . . . . . . . . . . 11 subringAlg RRfld subringAlg RRfld
3938fveq2d 5869 . . . . . . . . . 10 subringAlg RRfld subringAlg RRfld
4035, 36, 393eqtr4rd 2519 . . . . . . . . 9 subringAlg RRfld
4140adantl 466 . . . . . . . 8 subringAlg RRfld
4241ixpeq2dva 7484 . . . . . . 7 subringAlg RRfld
43 reex 9582 . . . . . . . 8
44 ixpconstg 7478 . . . . . . . 8
4543, 44mpan2 671 . . . . . . 7
4628, 42, 453eqtrd 2512 . . . . . 6 RRflds subringAlg RRfld
47 remulr 18430 . . . . . . . . . . 11 RRfld
4838, 34sraip 17624 . . . . . . . . . . 11 RRfld subringAlg RRfld
4947, 48syl5req 2521 . . . . . . . . . 10 subringAlg RRfld
5049oveqd 6300 . . . . . . . . 9 subringAlg RRfld
5150mpteq2ia 4529 . . . . . . . 8 subringAlg RRfld
5251a1i 11 . . . . . . 7 subringAlg RRfld
5352oveq2d 6299 . . . . . 6 RRfld g subringAlg RRfld RRfld g
5446, 46, 53mpt2eq123dv 6342 . . . . 5 RRflds subringAlg RRfld RRflds subringAlg RRfld RRfld g subringAlg RRfld RRfld g
5527, 54eqtrd 2508 . . . 4 RRflds subringAlg RRfld RRfld g
5613, 55syl5eqr 2522 . . 3 RRflds subringAlg RRflds RRfld g
577, 56syl5eqr 2522 . 2 toCHilRRflds subringAlg RRflds RRfld g
584, 57eqtr2d 2509 1 RRfld g
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1379   wcel 1767   wne 2662  cvv 3113   wss 3476  c0 3785  csn 4027   cmpt 4505   cxp 4997   cdm 4999  cfv 5587  (class class class)co 6283   cmpt2 6285   cmap 7420  cixp 7469  cr 9490   cmul 9496  cbs 14489   ↾s cress 14490  cmulr 14555  cip 14559   g cgsu 14695  scprds 14700  Fieldcfield 17192  subringAlg csra 17609  RRfldcrefld 18423  toCHilctch 21365  ℝ^crrx 21566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-rp 11220  df-fz 11672  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-0g 14696  df-prds 14702  df-pws 14704  df-mnd 15731  df-grp 15864  df-minusg 15865  df-subg 16000  df-cmn 16603  df-mgp 16941  df-ur 16953  df-rng 16997  df-cring 16998  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-dvr 17128  df-drng 17193  df-field 17194  df-subrg 17222  df-sra 17613  df-rgmod 17614  df-cnfld 18208  df-refld 18424  df-dsmm 18546  df-frlm 18561  df-tng 20856  df-tch 21367  df-rrx 21568 This theorem is referenced by:  rrxnm  21574
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