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Theorem rrxip 21688
 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 21687 . . 3 toCHilRRflds subringAlg RRflds
43fveq2d 5856 . 2 toCHilRRflds subringAlg RRflds
5 eqid 2441 . . . 4 toCHilRRflds subringAlg RRflds toCHilRRflds subringAlg RRflds
6 eqid 2441 . . . 4 RRflds subringAlg RRflds RRflds subringAlg RRflds
75, 6tchip 21534 . . 3 RRflds subringAlg RRflds toCHilRRflds subringAlg RRflds
8 fvex 5862 . . . . . 6
92, 8eqeltri 2525 . . . . 5
10 eqid 2441 . . . . . 6 RRflds subringAlg RRflds RRflds subringAlg RRflds
11 eqid 2441 . . . . . 6 RRflds subringAlg RRfld RRflds subringAlg RRfld
1210, 11ressip 14649 . . . . 5 RRflds subringAlg RRfld RRflds subringAlg RRflds
139, 12ax-mp 5 . . . 4 RRflds subringAlg RRfld RRflds subringAlg RRflds
14 eqid 2441 . . . . . 6 RRflds subringAlg RRfld RRflds subringAlg RRfld
15 refld 18522 . . . . . . 7 RRfld Field
1615a1i 11 . . . . . 6 RRfld Field
17 snex 4674 . . . . . . 7 subringAlg RRfld
18 xpexg 6583 . . . . . . 7 subringAlg RRfld subringAlg RRfld
1917, 18mpan2 671 . . . . . 6 subringAlg RRfld
20 eqid 2441 . . . . . 6 RRflds subringAlg RRfld RRflds subringAlg RRfld
21 fvex 5862 . . . . . . . . 9 subringAlg RRfld
2221snnz 4129 . . . . . . . 8 subringAlg RRfld
23 dmxp 5207 . . . . . . . 8 subringAlg RRfld subringAlg RRfld
2422, 23ax-mp 5 . . . . . . 7 subringAlg RRfld
2524a1i 11 . . . . . 6 subringAlg RRfld
2614, 16, 19, 20, 25, 11prdsip 14730 . . . . 5 RRflds subringAlg RRfld RRflds subringAlg RRfld RRflds subringAlg RRfld RRfld g subringAlg RRfld
2714, 16, 19, 20, 25prdsbas 14726 . . . . . . 7 RRflds subringAlg RRfld subringAlg RRfld
28 eqidd 2442 . . . . . . . . . . 11 subringAlg RRfld subringAlg RRfld
29 rebase 18509 . . . . . . . . . . . . 13 RRfld
3029eqimssi 3540 . . . . . . . . . . . 12 RRfld
3130a1i 11 . . . . . . . . . . 11 RRfld
3228, 31srabase 17692 . . . . . . . . . 10 RRfld subringAlg RRfld
3329a1i 11 . . . . . . . . . 10 RRfld
3421fvconst2 6107 . . . . . . . . . . 11 subringAlg RRfld subringAlg RRfld
3534fveq2d 5856 . . . . . . . . . 10 subringAlg RRfld subringAlg RRfld
3632, 33, 353eqtr4rd 2493 . . . . . . . . 9 subringAlg RRfld
3736adantl 466 . . . . . . . 8 subringAlg RRfld
3837ixpeq2dva 7482 . . . . . . 7 subringAlg RRfld
39 reex 9581 . . . . . . . 8
40 ixpconstg 7476 . . . . . . . 8
4139, 40mpan2 671 . . . . . . 7
4227, 38, 413eqtrd 2486 . . . . . 6 RRflds subringAlg RRfld
43 remulr 18514 . . . . . . . . . . 11 RRfld
4434, 31sraip 17697 . . . . . . . . . . 11 RRfld subringAlg RRfld
4543, 44syl5req 2495 . . . . . . . . . 10 subringAlg RRfld
4645oveqd 6294 . . . . . . . . 9 subringAlg RRfld
4746mpteq2ia 4515 . . . . . . . 8 subringAlg RRfld
4847a1i 11 . . . . . . 7 subringAlg RRfld
4948oveq2d 6293 . . . . . 6 RRfld g subringAlg RRfld RRfld g
5042, 42, 49mpt2eq123dv 6340 . . . . 5 RRflds subringAlg RRfld RRflds subringAlg RRfld RRfld g subringAlg RRfld RRfld g
5126, 50eqtrd 2482 . . . 4 RRflds subringAlg RRfld RRfld g
5213, 51syl5eqr 2496 . . 3 RRflds subringAlg RRflds RRfld g
537, 52syl5eqr 2496 . 2 toCHilRRflds subringAlg RRflds RRfld g
544, 53eqtr2d 2483 1 RRfld g
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1381   wcel 1802   wne 2636  cvv 3093   wss 3458  c0 3767  csn 4010   cmpt 4491   cxp 4983   cdm 4985  cfv 5574  (class class class)co 6277   cmpt2 6279   cmap 7418  cixp 7467  cr 9489   cmul 9495  cbs 14504   ↾s cress 14505  cmulr 14570  cip 14574   g cgsu 14710  scprds 14715  Fieldcfield 17265  subringAlg csra 17682  RRfldcrefld 18507  toCHilctch 21480  ℝ^crrx 21681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-rp 11225  df-fz 11677  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-0g 14711  df-prds 14717  df-pws 14719  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-subg 16067  df-cmn 16669  df-mgp 17010  df-ur 17022  df-ring 17068  df-cring 17069  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-dvr 17200  df-drng 17266  df-field 17267  df-subrg 17295  df-sra 17686  df-rgmod 17687  df-cnfld 18289  df-refld 18508  df-dsmm 18630  df-frlm 18645  df-tng 20971  df-tch 21482  df-rrx 21683 This theorem is referenced by:  rrxnm  21689
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