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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 28851 | . . . . 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 28655 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) ∈ 𝑋) |
7 | 2, 3, 3, 6 | mp3an 1463 | . . . . 5 ⊢ (𝐴𝐺𝐴) ∈ 𝑋 |
8 | ip2i.9 | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
9 | ip1i.7 | . . . . . 6 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
10 | 4, 9 | dipcl 28747 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ) |
11 | 2, 7, 8, 10 | mp3an 1463 | . . . 4 ⊢ ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ |
12 | 11 | addid1i 10984 | . . 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 28686 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈)) |
16 | 2, 3, 15 | mp2an 692 | . . . . . 6 ⊢ (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈) |
17 | 16 | oveq1i 7201 | . . . . 5 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵) |
18 | 4, 14, 9 | dip0l 28753 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0) |
19 | 2, 8, 18 | mp2an 692 | . . . . 5 ⊢ ((0vec‘𝑈)𝑃𝐵) = 0 |
20 | 17, 19 | eqtri 2759 | . . . 4 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = 0 |
21 | 20 | oveq2i 7202 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (((𝐴𝐺𝐴)𝑃𝐵) + 0) |
22 | df-2 11858 | . . . . . 6 ⊢ 2 = (1 + 1) | |
23 | 22 | oveq1i 7201 | . . . . 5 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴) |
24 | ax-1cn 10752 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
25 | 24, 24, 3 | 3pm3.2i 1341 | . . . . . . 7 ⊢ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) |
26 | 4, 5, 13 | nvdir 28666 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴))) |
27 | 2, 25, 26 | mp2an 692 | . . . . . 6 ⊢ ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)) |
28 | 4, 13 | nvsid 28662 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
29 | 2, 3, 28 | mp2an 692 | . . . . . . 7 ⊢ (1𝑆𝐴) = 𝐴 |
30 | 29, 29 | oveq12i 7203 | . . . . . 6 ⊢ ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴) |
31 | 27, 30 | eqtri 2759 | . . . . 5 ⊢ ((1 + 1)𝑆𝐴) = (𝐴𝐺𝐴) |
32 | 23, 31 | eqtri 2759 | . . . 4 ⊢ (2𝑆𝐴) = (𝐴𝐺𝐴) |
33 | 32 | oveq1i 7201 | . . 3 ⊢ ((2𝑆𝐴)𝑃𝐵) = ((𝐴𝐺𝐴)𝑃𝐵) |
34 | 12, 21, 33 | 3eqtr4ri 2770 | . 2 ⊢ ((2𝑆𝐴)𝑃𝐵) = (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) |
35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 28862 | . 2 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (2 · (𝐴𝑃𝐵)) |
36 | 34, 35 | eqtri 2759 | 1 ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 -cneg 11028 2c2 11850 NrmCVeccnv 28619 +𝑣 cpv 28620 BaseSetcba 28621 ·𝑠OLD cns 28622 0veccn0v 28623 ·𝑖OLDcdip 28735 CPreHilOLDccphlo 28847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 df-grpo 28528 df-gid 28529 df-ginv 28530 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-nmcv 28635 df-dip 28736 df-ph 28848 |
This theorem is referenced by: ipdirilem 28864 |
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