Theorem ipdiri 29772

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 7364	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝐺𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵))
2	1	oveq1d 7372	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶))
3		oveq1 7364	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝑃𝐶) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶))
4	3	oveq1d 7372	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶)))
5	2, 4	eqeq12d 2752	. 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 7365	. . . 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 7372	. . 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 7364	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶))
9	8	oveq2d 7373	. . 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 2752	. 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 7365	. . 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 7365	. . . 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 7365	. . . 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 7375	. . 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 2752	. 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 2736	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
22	16, 21, 20	elimph 29762	. . 3 ⊢ if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) ∈ 𝑋
23	16, 21, 20	elimph 29762	. . 3 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) ∈ 𝑋
24	16, 21, 20	elimph 29762	. . 3 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) ∈ 𝑋
25	16, 17, 18, 19, 20, 22, 23, 24	ipdirilem 29771	. 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 4546	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ifcif 4486  ‘cfv 6496  (class class class)co 7357   + caddc 11054   +_𝑣 cpv 29527  BaseSetcba 29528   ·_𝑠OLD cns 29529  0_veccn0v 29530  ·_𝑖OLDcdip 29642  CPreHil_OLDccphlo 29754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-grpo 29435  df-gid 29436  df-ginv 29437  df-ablo 29487  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-nmcv 29542  df-dip 29643  df-ph 29755

This theorem is referenced by:  ipasslem1  29773  ipasslem2  29774  ipasslem11  29782  dipdir  29784