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Theorem axhvaddid-zf 28690
Description: Derive axiom ax-hvaddid 28708 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 𝑈 = ⟨⟨ + , · ⟩, norm
axhil.2 𝑈 ∈ CHilOLD
Assertion
Ref Expression
axhvaddid-zf (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)

Proof of Theorem axhvaddid-zf
StepHypRef Expression
1 axhil.2 . 2 𝑈 ∈ CHilOLD
2 df-hba 28673 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6666 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2844 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 28596 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hva 28678 . . 3 + = ( +𝑣𝑈)
8 df-h0v 28674 . . . 4 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
93fveq2i 6666 . . . 4 (0vec𝑈) = (0vec‘⟨⟨ + , · ⟩, norm⟩)
108, 9eqtr4i 2844 . . 3 0 = (0vec𝑈)
115, 7, 10hladdid 28607 . 2 ((𝑈 ∈ CHilOLD𝐴 ∈ ℋ) → (𝐴 + 0) = 𝐴)
121, 11mpan 686 1 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cop 4563  cfv 6348  (class class class)co 7145  BaseSetcba 28290  0veccn0v 28292  CHilOLDchlo 28589  chba 28623   + cva 28624   · csm 28625  normcno 28627  0c0v 28628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gid 28198  df-ablo 28249  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-nmcv 28304  df-cbn 28567  df-hlo 28590  df-hba 28673  df-h0v 28674
This theorem is referenced by: (None)
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