Theorem ipdiri 30989

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 7397	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝐺𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵))
2	1	oveq1d 7405	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝐺𝐵)𝑃𝐶))
3		oveq1 7397	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝑃𝐶) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶))
4	3	oveq1d 7405	. . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) = ((if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐶) + (𝐵𝑃𝐶)))
5	2, 4	eqeq12d 2777	. 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 7398	. . . 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 7405	. . 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 7397	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))𝑃𝐶))
9	8	oveq2d 7406	. . 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 2777	. 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 7398	. . 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 7398	. . . 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 7398	. . . 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 7408	. . 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 2777	. 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 2761	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
22	16, 21, 20	elimph 30979	. . 3 ⊢ if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) ∈ 𝑋
23	16, 21, 20	elimph 30979	. . 3 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) ∈ 𝑋
24	16, 21, 20	elimph 30979	. . 3 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0vec‘𝑈)) ∈ 𝑋
25	16, 17, 18, 19, 20, 22, 23, 24	ipdirilem 30988	. 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 4538	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ifcif 4477  ‘cfv 6515  (class class class)co 7390   + caddc 11069   +_𝑣 cpv 30744  BaseSetcba 30745   ·_𝑠OLD cns 30746  0_veccn0v 30747  ·_𝑖OLDcdip 30859  CPreHil_OLDccphlo 30971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-inf2 9589  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143  ax-pre-sup 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-sup 9381  df-oi 9451  df-card 9890  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-2 12273  df-3 12274  df-4 12275  df-n0 12475  df-z 12562  df-uz 12833  df-rp 12987  df-fz 13506  df-fzo 13653  df-seq 14008  df-exp 14068  df-hash 14337  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15505  df-sum 15704  df-grpo 30652  df-gid 30653  df-ginv 30654  df-ablo 30704  df-vc 30718  df-nv 30751  df-va 30754  df-ba 30755  df-sm 30756  df-0v 30757  df-nmcv 30759  df-dip 30860  df-ph 30972

This theorem is referenced by:  ipasslem1  30990  ipasslem2  30991  ipasslem11  30999  dipdir  31001