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Theorem 0lno 28582
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 2824 . . 3 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 eqid 2824 . . 3 (BaseSet‘𝑊) = (BaseSet‘𝑊)
3 0lno.0 . . 3 𝑍 = (𝑈 0op 𝑊)
41, 2, 30oo 28581 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
5 simplll 774 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑈 ∈ NrmCVec)
6 simpllr 775 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑊 ∈ NrmCVec)
7 simplr 768 . . . . . . . 8 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑥 ∈ ℂ)
8 simprl 770 . . . . . . . 8 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑦 ∈ (BaseSet‘𝑈))
9 eqid 2824 . . . . . . . . 9 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
101, 9nvscl 28418 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
115, 7, 8, 10syl3anc 1368 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
12 simprr 772 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑧 ∈ (BaseSet‘𝑈))
13 eqid 2824 . . . . . . . 8 ( +𝑣𝑈) = ( +𝑣𝑈)
141, 13nvgcl 28412 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
155, 11, 12, 14syl3anc 1368 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
16 eqid 2824 . . . . . . 7 (0vec𝑊) = (0vec𝑊)
171, 16, 30oval 28580 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈)) → (𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = (0vec𝑊))
185, 6, 15, 17syl3anc 1368 . . . . 5 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = (0vec𝑊))
191, 16, 30oval 28580 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑍𝑦) = (0vec𝑊))
205, 6, 8, 19syl3anc 1368 . . . . . . . 8 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍𝑦) = (0vec𝑊))
2120oveq2d 7167 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD𝑊)(𝑍𝑦)) = (𝑥( ·𝑠OLD𝑊)(0vec𝑊)))
221, 16, 30oval 28580 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍𝑧) = (0vec𝑊))
235, 6, 12, 22syl3anc 1368 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍𝑧) = (0vec𝑊))
2421, 23oveq12d 7169 . . . . . 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 2824 . . . . . . . . 9 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2625, 16nvsz 28430 . . . . . . . 8 ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ) → (𝑥( ·𝑠OLD𝑊)(0vec𝑊)) = (0vec𝑊))
276, 7, 26syl2anc 587 . . . . . . 7 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD𝑊)(0vec𝑊)) = (0vec𝑊))
2827oveq1d 7166 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑊)(0vec𝑊))( +𝑣𝑊)(0vec𝑊)) = ((0vec𝑊)( +𝑣𝑊)(0vec𝑊)))
292, 16nvzcl 28426 . . . . . . 7 (𝑊 ∈ NrmCVec → (0vec𝑊) ∈ (BaseSet‘𝑊))
30 eqid 2824 . . . . . . . 8 ( +𝑣𝑊) = ( +𝑣𝑊)
312, 30, 16nv0rid 28427 . . . . . . 7 ((𝑊 ∈ NrmCVec ∧ (0vec𝑊) ∈ (BaseSet‘𝑊)) → ((0vec𝑊)( +𝑣𝑊)(0vec𝑊)) = (0vec𝑊))
326, 29, 31syl2anc2 588 . . . . . 6 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((0vec𝑊)( +𝑣𝑊)(0vec𝑊)) = (0vec𝑊))
3324, 28, 323eqtrd 2863 . . . . 5 ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD𝑊)(𝑍𝑦))( +𝑣𝑊)(𝑍𝑧)) = (0vec𝑊))
3418, 33eqtr4d 2862 . . . 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 3186 . . 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 3177 . 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 28545 . 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 712 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wf 6341  cfv 6345  (class class class)co 7151  cc 10535  NrmCVeccnv 28376   +𝑣 cpv 28377  BaseSetcba 28378   ·𝑠OLD cns 28379  0veccn0v 28380   LnOp clno 28532   0op c0o 28535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-po 5462  df-so 5463  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-pnf 10677  df-mnf 10678  df-ltxr 10680  df-grpo 28285  df-gid 28286  df-ginv 28287  df-ablo 28337  df-vc 28351  df-nv 28384  df-va 28387  df-ba 28388  df-sm 28389  df-0v 28390  df-nmcv 28392  df-lno 28536  df-0o 28539
This theorem is referenced by:  0blo  28584  nmlno0i  28586  blocn  28599
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