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Theorem 0lno 30819
Description: The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
0lno.0 𝑍 = (𝑈 0op 𝑊)
0lno.7 𝐿 = (𝑈 LnOp 𝑊)
Assertion
Ref Expression
0lno ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍𝐿)

Proof of Theorem 0lno
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 eqid 2735 . . 3 (BaseSet‘𝑊) = (BaseSet‘𝑊)
3 0lno.0 . . 3 𝑍 = (𝑈 0op 𝑊)
41, 2, 30oo 30818 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
5 simplll 775 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑈 ∈ NrmCVec)
6 simpllr 776 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑊 ∈ NrmCVec)
7 simplr 769 . . . . . . . 8 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑥 ∈ ℂ)
8 simprl 771 . . . . . . . 8 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑦 ∈ (BaseSet‘𝑈))
9 eqid 2735 . . . . . . . . 9 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
101, 9nvscl 30655 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
115, 7, 8, 10syl3anc 1370 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
12 simprr 773 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑧 ∈ (BaseSet‘𝑈))
13 eqid 2735 . . . . . . . 8 ( +𝑣𝑈) = ( +𝑣𝑈)
141, 13nvgcl 30649 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
155, 11, 12, 14syl3anc 1370 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
16 eqid 2735 . . . . . . 7 (0vec𝑊) = (0vec𝑊)
171, 16, 30oval 30817 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈)) → (𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = (0vec𝑊))
185, 6, 15, 17syl3anc 1370 . . . . 5 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = (0vec𝑊))
191, 16, 30oval 30817 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑍𝑦) = (0vec𝑊))
205, 6, 8, 19syl3anc 1370 . . . . . . . 8 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍𝑦) = (0vec𝑊))
2120oveq2d 7447 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD𝑊)(𝑍𝑦)) = (𝑥( ·𝑠OLD𝑊)(0vec𝑊)))
221, 16, 30oval 30817 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍𝑧) = (0vec𝑊))
235, 6, 12, 22syl3anc 1370 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍𝑧) = (0vec𝑊))
2421, 23oveq12d 7449 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)) = ((𝑥( ·𝑠OLD𝑊)(0vec𝑊))( +𝑣𝑊)(0vec𝑊)))
25 eqid 2735 . . . . . . . . 9 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2625, 16nvsz 30667 . . . . . . . 8 ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ) → (𝑥( ·𝑠OLD𝑊)(0vec𝑊)) = (0vec𝑊))
276, 7, 26syl2anc 584 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD𝑊)(0vec𝑊)) = (0vec𝑊))
2827oveq1d 7446 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑊)(0vec𝑊))( +𝑣𝑊)(0vec𝑊)) = ((0vec𝑊)( +𝑣𝑊)(0vec𝑊)))
292, 16nvzcl 30663 . . . . . . 7 (𝑊 ∈ NrmCVec → (0vec𝑊) ∈ (BaseSet‘𝑊))
30 eqid 2735 . . . . . . . 8 ( +𝑣𝑊) = ( +𝑣𝑊)
312, 30, 16nv0rid 30664 . . . . . . 7 ((𝑊 ∈ NrmCVec ∧ (0vec𝑊) ∈ (BaseSet‘𝑊)) → ((0vec𝑊)( +𝑣𝑊)(0vec𝑊)) = (0vec𝑊))
326, 29, 31syl2anc2 585 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((0vec𝑊)( +𝑣𝑊)(0vec𝑊)) = (0vec𝑊))
3324, 28, 323eqtrd 2779 . . . . 5 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)) = (0vec𝑊))
3418, 33eqtr4d 2778 . . . 4 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)))
3534ralrimivva 3200 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) → ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)(𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)))
3635ralrimiva 3144 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)(𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)))
37 0lno.7 . . 3 𝐿 = (𝑈 LnOp 𝑊)
381, 2, 13, 30, 9, 25, 37islno 30782 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑍𝐿 ↔ (𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)(𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)))))
394, 36, 38mpbir2and 713 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  NrmCVeccnv 30613   +𝑣 cpv 30614  BaseSetcba 30615   ·𝑠OLD cns 30616  0veccn0v 30617   LnOp clno 30769   0op c0o 30772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-ltxr 11298  df-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-lno 30773  df-0o 30776
This theorem is referenced by:  0blo  30821  nmlno0i  30823  blocn  30836
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