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Theorem ipdiri 27069
 Description: Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1 𝑋 = (BaseSet‘𝑈)
ip1i.2 𝐺 = ( +𝑣𝑈)
ip1i.4 𝑆 = ( ·𝑠OLD𝑈)
ip1i.7 𝑃 = (·𝑖OLD𝑈)
ip1i.9 𝑈 ∈ CPreHilOLD
Assertion
Ref Expression
ipdiri ((𝐴𝑋𝐵𝑋𝐶𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Proof of Theorem ipdiri
StepHypRef Expression
1 oveq1 6556 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, (0vec𝑈)) → (𝐴𝐺𝐵) = (if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺𝐵))
21oveq1d 6564 . . 3 (𝐴 = if(𝐴𝑋, 𝐴, (0vec𝑈)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺𝐵)𝑃𝐶))
3 oveq1 6556 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, (0vec𝑈)) → (𝐴𝑃𝐶) = (if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶))
43oveq1d 6564 . . 3 (𝐴 = if(𝐴𝑋, 𝐴, (0vec𝑈)) → ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (𝐵𝑃𝐶)))
52, 4eqeq12d 2625 . 2 (𝐴 = if(𝐴𝑋, 𝐴, (0vec𝑈)) → (((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) ↔ ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺𝐵)𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (𝐵𝑃𝐶))))
6 oveq2 6557 . . . 4 (𝐵 = if(𝐵𝑋, 𝐵, (0vec𝑈)) → (if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺𝐵) = (if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈))))
76oveq1d 6564 . . 3 (𝐵 = if(𝐵𝑋, 𝐵, (0vec𝑈)) → ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺𝐵)𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃𝐶))
8 oveq1 6556 . . . 4 (𝐵 = if(𝐵𝑋, 𝐵, (0vec𝑈)) → (𝐵𝑃𝐶) = (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃𝐶))
98oveq2d 6565 . . 3 (𝐵 = if(𝐵𝑋, 𝐵, (0vec𝑈)) → ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃𝐶)))
107, 9eqeq12d 2625 . 2 (𝐵 = if(𝐵𝑋, 𝐵, (0vec𝑈)) → (((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺𝐵)𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (𝐵𝑃𝐶)) ↔ ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃𝐶))))
11 oveq2 6557 . . 3 (𝐶 = if(𝐶𝑋, 𝐶, (0vec𝑈)) → ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))))
12 oveq2 6557 . . . 4 (𝐶 = if(𝐶𝑋, 𝐶, (0vec𝑈)) → (if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) = (if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))))
13 oveq2 6557 . . . 4 (𝐶 = if(𝐶𝑋, 𝐶, (0vec𝑈)) → (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃𝐶) = (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))))
1412, 13oveq12d 6567 . . 3 (𝐶 = if(𝐶𝑋, 𝐶, (0vec𝑈)) → ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃𝐶)) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈)))))
1511, 14eqeq12d 2625 . 2 (𝐶 = if(𝐶𝑋, 𝐶, (0vec𝑈)) → (((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃𝐶) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃𝐶) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃𝐶)) ↔ ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))))))
16 ip1i.1 . . 3 𝑋 = (BaseSet‘𝑈)
17 ip1i.2 . . 3 𝐺 = ( +𝑣𝑈)
18 ip1i.4 . . 3 𝑆 = ( ·𝑠OLD𝑈)
19 ip1i.7 . . 3 𝑃 = (·𝑖OLD𝑈)
20 ip1i.9 . . 3 𝑈 ∈ CPreHilOLD
21 eqid 2610 . . . 4 (0vec𝑈) = (0vec𝑈)
2216, 21, 20elimph 27059 . . 3 if(𝐴𝑋, 𝐴, (0vec𝑈)) ∈ 𝑋
2316, 21, 20elimph 27059 . . 3 if(𝐵𝑋, 𝐵, (0vec𝑈)) ∈ 𝑋
2416, 21, 20elimph 27059 . . 3 if(𝐶𝑋, 𝐶, (0vec𝑈)) ∈ 𝑋
2516, 17, 18, 19, 20, 22, 23, 24ipdirilem 27068 . 2 ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝐺if(𝐵𝑋, 𝐵, (0vec𝑈)))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))) = ((if(𝐴𝑋, 𝐴, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))) + (if(𝐵𝑋, 𝐵, (0vec𝑈))𝑃if(𝐶𝑋, 𝐶, (0vec𝑈))))
265, 10, 15, 25dedth3h 4091 1 ((𝐴𝑋𝐵𝑋𝐶𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ifcif 4036  ‘cfv 5804  (class class class)co 6549   + caddc 9818   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826  0veccn0v 26827  ·𝑖OLDcdip 26939  CPreHilOLDccphlo 27051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-dip 26940  df-ph 27052 This theorem is referenced by:  ipasslem1  27070  ipasslem2  27071  ipasslem11  27079  dipdir  27081
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