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Mirrors > Home > MPE Home > Th. List > ip2i | Structured version Visualization version GIF version |
Description: Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
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 |
ip2i.8 | ⊢ 𝐴 ∈ 𝑋 |
ip2i.9 | ⊢ 𝐵 ∈ 𝑋 |
Ref | Expression |
---|---|
ip2i | ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ip1i.9 | . . . . . 6 ⊢ 𝑈 ∈ CPreHilOLD | |
2 | 1 | phnvi 29644 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
3 | ip2i.8 | . . . . . 6 ⊢ 𝐴 ∈ 𝑋 | |
4 | ip1i.1 | . . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈) | |
5 | ip1i.2 | . . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
6 | 4, 5 | nvgcl 29448 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) ∈ 𝑋) |
7 | 2, 3, 3, 6 | mp3an 1461 | . . . . 5 ⊢ (𝐴𝐺𝐴) ∈ 𝑋 |
8 | ip2i.9 | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
9 | ip1i.7 | . . . . . 6 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
10 | 4, 9 | dipcl 29540 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ) |
11 | 2, 7, 8, 10 | mp3an 1461 | . . . 4 ⊢ ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ |
12 | 11 | addid1i 11338 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + 0) = ((𝐴𝐺𝐴)𝑃𝐵) |
13 | ip1i.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
14 | eqid 2736 | . . . . . . . 8 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
15 | 4, 5, 13, 14 | nvrinv 29479 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈)) |
16 | 2, 3, 15 | mp2an 690 | . . . . . 6 ⊢ (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈) |
17 | 16 | oveq1i 7363 | . . . . 5 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵) |
18 | 4, 14, 9 | dip0l 29546 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0) |
19 | 2, 8, 18 | mp2an 690 | . . . . 5 ⊢ ((0vec‘𝑈)𝑃𝐵) = 0 |
20 | 17, 19 | eqtri 2764 | . . . 4 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = 0 |
21 | 20 | oveq2i 7364 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (((𝐴𝐺𝐴)𝑃𝐵) + 0) |
22 | df-2 12212 | . . . . . 6 ⊢ 2 = (1 + 1) | |
23 | 22 | oveq1i 7363 | . . . . 5 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴) |
24 | ax-1cn 11105 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
25 | 24, 24, 3 | 3pm3.2i 1339 | . . . . . . 7 ⊢ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) |
26 | 4, 5, 13 | nvdir 29459 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
27 | 2, 25, 26 | mp2an 690 | . . . . . 6 ⊢ ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)) |
28 | 4, 13 | nvsid 29455 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
29 | 2, 3, 28 | mp2an 690 | . . . . . . 7 ⊢ (1𝑆𝐴) = 𝐴 |
30 | 29, 29 | oveq12i 7365 | . . . . . 6 ⊢ ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴) |
31 | 27, 30 | eqtri 2764 | . . . . 5 ⊢ ((1 + 1)𝑆𝐴) = (𝐴𝐺𝐴) |
32 | 23, 31 | eqtri 2764 | . . . 4 ⊢ (2𝑆𝐴) = (𝐴𝐺𝐴) |
33 | 32 | oveq1i 7363 | . . 3 ⊢ ((2𝑆𝐴)𝑃𝐵) = ((𝐴𝐺𝐴)𝑃𝐵) |
34 | 12, 21, 33 | 3eqtr4ri 2775 | . 2 ⊢ ((2𝑆𝐴)𝑃𝐵) = (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) |
35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 29655 | . 2 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (2 · (𝐴𝑃𝐵)) |
36 | 34, 35 | eqtri 2764 | 1 ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 ℂcc 11045 0cc0 11047 1c1 11048 + caddc 11050 · cmul 11052 -cneg 11382 2c2 12204 NrmCVeccnv 29412 +𝑣 cpv 29413 BaseSetcba 29414 ·𝑠OLD cns 29415 0veccn0v 29416 ·𝑖OLDcdip 29528 CPreHilOLDccphlo 29640 |
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 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-inf2 9573 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
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 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9374 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-fz 13417 df-fzo 13560 df-seq 13899 df-exp 13960 df-hash 14223 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-clim 15362 df-sum 15563 df-grpo 29321 df-gid 29322 df-ginv 29323 df-ablo 29373 df-vc 29387 df-nv 29420 df-va 29423 df-ba 29424 df-sm 29425 df-0v 29426 df-nmcv 29428 df-dip 29529 df-ph 29641 |
This theorem is referenced by: ipdirilem 29657 |
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