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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 30835 | . . . . 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 30639 | . . . . . 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 30731 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ) |
| 11 | 2, 7, 8, 10 | mp3an 1463 | . . . 4 ⊢ ((𝐴𝐺𝐴)𝑃𝐵) ∈ ℂ |
| 12 | 11 | addridi 11448 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + 0) = ((𝐴𝐺𝐴)𝑃𝐵) |
| 13 | ip1i.4 | . . . . . . . 8 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 14 | eqid 2737 | . . . . . . . 8 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
| 15 | 4, 5, 13, 14 | nvrinv 30670 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈)) |
| 16 | 2, 3, 15 | mp2an 692 | . . . . . 6 ⊢ (𝐴𝐺(-1𝑆𝐴)) = (0vec‘𝑈) |
| 17 | 16 | oveq1i 7441 | . . . . 5 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = ((0vec‘𝑈)𝑃𝐵) |
| 18 | 4, 14, 9 | dip0l 30737 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((0vec‘𝑈)𝑃𝐵) = 0) |
| 19 | 2, 8, 18 | mp2an 692 | . . . . 5 ⊢ ((0vec‘𝑈)𝑃𝐵) = 0 |
| 20 | 17, 19 | eqtri 2765 | . . . 4 ⊢ ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵) = 0 |
| 21 | 20 | oveq2i 7442 | . . 3 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (((𝐴𝐺𝐴)𝑃𝐵) + 0) |
| 22 | df-2 12329 | . . . . . 6 ⊢ 2 = (1 + 1) | |
| 23 | 22 | oveq1i 7441 | . . . . 5 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴) |
| 24 | ax-1cn 11213 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
| 25 | 24, 24, 3 | 3pm3.2i 1340 | . . . . . . 7 ⊢ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) |
| 26 | 4, 5, 13 | nvdir 30650 | . . . . . . 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 30646 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
| 29 | 2, 3, 28 | mp2an 692 | . . . . . . 7 ⊢ (1𝑆𝐴) = 𝐴 |
| 30 | 29, 29 | oveq12i 7443 | . . . . . 6 ⊢ ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴) |
| 31 | 27, 30 | eqtri 2765 | . . . . 5 ⊢ ((1 + 1)𝑆𝐴) = (𝐴𝐺𝐴) |
| 32 | 23, 31 | eqtri 2765 | . . . 4 ⊢ (2𝑆𝐴) = (𝐴𝐺𝐴) |
| 33 | 32 | oveq1i 7441 | . . 3 ⊢ ((2𝑆𝐴)𝑃𝐵) = ((𝐴𝐺𝐴)𝑃𝐵) |
| 34 | 12, 21, 33 | 3eqtr4ri 2776 | . 2 ⊢ ((2𝑆𝐴)𝑃𝐵) = (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) |
| 35 | 4, 5, 13, 9, 1, 3, 3, 8 | ip1i 30846 | . 2 ⊢ (((𝐴𝐺𝐴)𝑃𝐵) + ((𝐴𝐺(-1𝑆𝐴))𝑃𝐵)) = (2 · (𝐴𝑃𝐵)) |
| 36 | 34, 35 | eqtri 2765 | 1 ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 -cneg 11493 2c2 12321 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 0veccn0v 30607 ·𝑖OLDcdip 30719 CPreHilOLDccphlo 30831 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 df-dip 30720 df-ph 30832 |
| This theorem is referenced by: ipdirilem 30848 |
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