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Theorem axhvaddid-zf 28744
Description: Derive axiom ax-hvaddid 28762 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 28727 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6645 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2846 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 28650 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hva 28732 . . 3 + = ( +𝑣𝑈)
8 df-h0v 28728 . . . 4 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
93fveq2i 6645 . . . 4 (0vec𝑈) = (0vec‘⟨⟨ + , · ⟩, norm⟩)
108, 9eqtr4i 2846 . . 3 0 = (0vec𝑈)
115, 7, 10hladdid 28661 . 2 ((𝑈 ∈ CHilOLD𝐴 ∈ ℋ) → (𝐴 + 0) = 𝐴)
121, 11mpan 688 1 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  cop 4545  cfv 6327  (class class class)co 7129  BaseSetcba 28344  0veccn0v 28346  CHilOLDchlo 28643  chba 28677   + cva 28678   · csm 28679  normcno 28681  0c0v 28682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-oprab 7133  df-1st 7663  df-2nd 7664  df-grpo 28251  df-gid 28252  df-ablo 28303  df-vc 28317  df-nv 28350  df-va 28353  df-ba 28354  df-sm 28355  df-0v 28356  df-nmcv 28358  df-cbn 28621  df-hlo 28644  df-hba 28727  df-h0v 28728
This theorem is referenced by: (None)
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