Theorem ipdiri 29192

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 7282	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝐺𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵))
2	1	oveq1d 7290	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶))
3		oveq1 7282	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝑃𝐶) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶))
4	3	oveq1d 7290	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶)))
5	2, 4	eqeq12d 2754	. 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 7283	. . . 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 7290	. . 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 7282	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶))
9	8	oveq2d 7291	. . 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 2754	. 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 7283	. . 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 7283	. . . 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 7283	. . . 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 7293	. . 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 2754	. 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 2738	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
22	16, 21, 20	elimph 29182	. . 3 ⊢ if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) ∈ 𝑋
23	16, 21, 20	elimph 29182	. . 3 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) ∈ 𝑋
24	16, 21, 20	elimph 29182	. . 3 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) ∈ 𝑋
25	16, 17, 18, 19, 20, 22, 23, 24	ipdirilem 29191	. 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 4519	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ifcif 4459  ‘cfv 6433  (class class class)co 7275   + caddc 10874   +_𝑣 cpv 28947  BaseSetcba 28948   ·_𝑠OLD cns 28949  0_veccn0v 28950  ·_𝑖OLDcdip 29062  CPreHil_OLDccphlo 29174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-grpo 28855  df-gid 28856  df-ginv 28857  df-ablo 28907  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-nmcv 28962  df-dip 29063  df-ph 29175

This theorem is referenced by:  ipasslem1  29193  ipasslem2  29194  ipasslem11  29202  dipdir  29204