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Theorem axhfvadd-zf 31057

Description: Derive Axiom ax-hfvadd 31075 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
axhil.1	⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
axhil.2	⊢ 𝑈 ∈ CHil_OLD

**Assertion**
Ref	Expression
axhfvadd-zf	⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ

**Proof of Theorem axhfvadd-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 31044	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6837	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2762	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 30967	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hva 31049	. . 3 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)
8	5, 7	hladdf 30974	. 2 ⊢ (𝑈 ∈ CHil_OLD → +_ℎ :( ℋ × ℋ)⟶ ℋ)
9	1, 8	ax-mp 5	1 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 ⟨cop 4586 × cxp 5622 ⟶wf 6488 ‘cfv 6492 BaseSetcba 30661 CHil_OLDchlo 30960 ℋchba 30994 +_ℎ cva 30995 ·_ℎ csm 30996 norm_ℎcno 30998

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-1st 7933 df-2nd 7934 df-grpo 30568 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-nmcv 30675 df-cbn 30938 df-hlo 30961 df-hba 31044

This theorem is referenced by: (None)

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