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Mirrors > Home > MPE Home > Th. List > ipval2lem3 | Structured version Visualization version GIF version |
Description: Lemma for ipval3 30596. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dipfval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
dipfval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
dipfval.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
dipfval.6 | ⊢ 𝑁 = (normCV‘𝑈) |
dipfval.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
Ref | Expression |
---|---|
ipval2lem3 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dipfval.1 | . . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | dipfval.4 | . . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
3 | 1, 2 | nvsid 30514 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1𝑆𝐵) = 𝐵) |
4 | 3 | oveq2d 7435 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(1𝑆𝐵)) = (𝐴𝐺𝐵)) |
5 | 4 | fveq2d 6900 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(1𝑆𝐵))) = (𝑁‘(𝐴𝐺𝐵))) |
6 | 5 | oveq1d 7434 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2)) |
7 | 6 | 3adant2 1128 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2)) |
8 | ax-1cn 11203 | . . 3 ⊢ 1 ∈ ℂ | |
9 | dipfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
10 | dipfval.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
11 | dipfval.7 | . . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
12 | 1, 9, 2, 10, 11 | ipval2lem2 30591 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 1 ∈ ℂ) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) ∈ ℝ) |
13 | 8, 12 | mpan2 689 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) ∈ ℝ) |
14 | 7, 13 | eqeltrrd 2826 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 ℝcr 11144 1c1 11146 2c2 12305 ↑cexp 14067 NrmCVeccnv 30471 +𝑣 cpv 30472 BaseSetcba 30473 ·𝑠OLD cns 30474 normCVcnmcv 30477 ·𝑖OLDcdip 30587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-seq 14008 df-exp 14068 df-grpo 30380 df-ablo 30432 df-vc 30446 df-nv 30479 df-va 30482 df-ba 30483 df-sm 30484 df-0v 30485 df-nmcv 30487 |
This theorem is referenced by: ipval2 30594 dipcj 30601 dip0r 30604 |
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