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Theorem axhvaddid-zf 28302
Description: Derive axiom ax-hvaddid 28320 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 28285 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6380 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2790 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 28207 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hva 28290 . . 3 + = ( +𝑣𝑈)
8 df-h0v 28286 . . . 4 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
93fveq2i 6380 . . . 4 (0vec𝑈) = (0vec‘⟨⟨ + , · ⟩, norm⟩)
108, 9eqtr4i 2790 . . 3 0 = (0vec𝑈)
115, 7, 10hladdid 28218 . 2 ((𝑈 ∈ CHilOLD𝐴 ∈ ℋ) → (𝐴 + 0) = 𝐴)
121, 11mpan 681 1 (𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  cop 4342  cfv 6070  (class class class)co 6844  BaseSetcba 27900  0veccn0v 27902  CHilOLDchlo 28200  chba 28235   + cva 28236   · csm 28237  normcno 28239  0c0v 28240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-1st 7368  df-2nd 7369  df-grpo 27807  df-gid 27808  df-ablo 27859  df-vc 27873  df-nv 27906  df-va 27909  df-ba 27910  df-sm 27911  df-0v 27912  df-nmcv 27914  df-cbn 28178  df-hlo 28201  df-hba 28285  df-h0v 28286
This theorem is referenced by: (None)
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