Theorem ipdiri 30812

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 = ( +𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHilOLD

Assertion

Ref	Expression
ipdiri	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Proof of Theorem ipdiri

Step	Hyp	Ref	Expression
1		oveq1 7359	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝐺𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵))
2	1	oveq1d 7367	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶))
3		oveq1 7359	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝑃𝐶) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶))
4	3	oveq1d 7367	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶)))
5	2, 4	eqeq12d 2749	. 2 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) ↔ ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶))))
6		oveq2 7360	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))))
7	6	oveq1d 7367	. . 3 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃𝐶))
8		oveq1 7359	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶))
9	8	oveq2d 7368	. . 3 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶)))
10	7, 9	eqeq12d 2749	. 2 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶)) ↔ ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶))))
11		oveq2 7360	. . 3 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) → ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))))
12		oveq2 7360	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) → (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))))
13		oveq2 7360	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) → (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))))
14	12, 13	oveq12d 7370	. . 3 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) → ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶)) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)))))
15	11, 14	eqeq12d 2749	. 2 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) → (((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶)) ↔ ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))))))
16		ip1i.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
17		ip1i.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
18		ip1i.4	. . 3 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
19		ip1i.7	. . 3 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
20		ip1i.9	. . 3 ⊢ 𝑈 ∈ CPreHilOLD
21		eqid 2733	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
22	16, 21, 20	elimph 30802	. . 3 ⊢ if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) ∈ 𝑋
23	16, 21, 20	elimph 30802	. . 3 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) ∈ 𝑋
24	16, 21, 20	elimph 30802	. . 3 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) ∈ 𝑋
25	16, 17, 18, 19, 20, 22, 23, 24	ipdirilem 30811	. 2 ⊢ ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))) + (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈))))
26	5, 10, 15, 25	dedth3h 4535	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ifcif 4474  ‘cfv 6486  (class class class)co 7352   + caddc 11016   +_𝑣 cpv 30567  BaseSetcba 30568   ·_𝑠OLD cns 30569  0_veccn0v 30570  ·_𝑖OLDcdip 30682  CPreHil_OLDccphlo 30794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-n0 12389  df-z 12476  df-uz 12739  df-rp 12893  df-fz 13410  df-fzo 13557  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-sum 15596  df-grpo 30475  df-gid 30476  df-ginv 30477  df-ablo 30527  df-vc 30541  df-nv 30574  df-va 30577  df-ba 30578  df-sm 30579  df-0v 30580  df-nmcv 30582  df-dip 30683  df-ph 30795

This theorem is referenced by:  ipasslem1  30813  ipasslem2  30814  ipasslem11  30822  dipdir  30824