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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 28595 | . . . . 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 28399 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) ∈ 𝑋) |
7 | 2, 3, 3, 6 | mp3an 1457 | . . . . 5 ⊢ (𝐴𝐺𝐴) ∈ 𝑋 |
8 | ip2i.9 | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
9 | ip1i.7 | . . . . . 6 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
10 | 4, 9 | dipcl 28491 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ) |
11 | 2, 7, 8, 10 | mp3an 1457 | . . . 4 ⊢ ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ |
12 | 11 | addid1i 10829 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + 0) = ((𝐴𝐺𝐴)𝑃𝐵) |
13 | ip1i.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
14 | eqid 2823 | . . . . . . . 8 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
15 | 4, 5, 13, 14 | nvrinv 28430 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈)) |
16 | 2, 3, 15 | mp2an 690 | . . . . . 6 ⊢ (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈) |
17 | 16 | oveq1i 7168 | . . . . 5 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵) |
18 | 4, 14, 9 | dip0l 28497 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0) |
19 | 2, 8, 18 | mp2an 690 | . . . . 5 ⊢ ((0vec‘𝑈)𝑃𝐵) = 0 |
20 | 17, 19 | eqtri 2846 | . . . 4 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = 0 |
21 | 20 | oveq2i 7169 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (((𝐴𝐺𝐴)𝑃𝐵) + 0) |
22 | df-2 11703 | . . . . . 6 ⊢ 2 = (1 + 1) | |
23 | 22 | oveq1i 7168 | . . . . 5 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴) |
24 | ax-1cn 10597 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
25 | 24, 24, 3 | 3pm3.2i 1335 | . . . . . . 7 ⊢ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) |
26 | 4, 5, 13 | nvdir 28410 | . . . . . . 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 28406 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
29 | 2, 3, 28 | mp2an 690 | . . . . . . 7 ⊢ (1𝑆𝐴) = 𝐴 |
30 | 29, 29 | oveq12i 7170 | . . . . . 6 ⊢ ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴) |
31 | 27, 30 | eqtri 2846 | . . . . 5 ⊢ ((1 + 1)𝑆𝐴) = (𝐴𝐺𝐴) |
32 | 23, 31 | eqtri 2846 | . . . 4 ⊢ (2𝑆𝐴) = (𝐴𝐺𝐴) |
33 | 32 | oveq1i 7168 | . . 3 ⊢ ((2𝑆𝐴)𝑃𝐵) = ((𝐴𝐺𝐴)𝑃𝐵) |
34 | 12, 21, 33 | 3eqtr4ri 2857 | . 2 ⊢ ((2𝑆𝐴)𝑃𝐵) = (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) |
35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 28606 | . 2 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (2 · (𝐴𝑃𝐵)) |
36 | 34, 35 | eqtri 2846 | 1 ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 -cneg 10873 2c2 11695 NrmCVeccnv 28363 +𝑣 cpv 28364 BaseSetcba 28365 ·𝑠OLD cns 28366 0veccn0v 28367 ·𝑖OLDcdip 28479 CPreHilOLDccphlo 28591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-grpo 28272 df-gid 28273 df-ginv 28274 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-nmcv 28379 df-dip 28480 df-ph 28592 |
This theorem is referenced by: ipdirilem 28608 |
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