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Mirrors > Home > HSE Home > Th. List > axhvaddid-zf | Structured version Visualization version GIF version |
Description: Derive axiom ax-hvaddid 28784 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axhil.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
axhil.2 | ⊢ 𝑈 ∈ CHilOLD |
Ref | Expression |
---|---|
axhvaddid-zf | ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axhil.2 | . 2 ⊢ 𝑈 ∈ CHilOLD | |
2 | df-hba 28749 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | axhil.1 | . . . . 5 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
4 | 3 | fveq2i 6676 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | 2, 4 | eqtr4i 2850 | . . 3 ⊢ ℋ = (BaseSet‘𝑈) |
6 | 1 | hlnvi 28672 | . . . 4 ⊢ 𝑈 ∈ NrmCVec |
7 | 3, 6 | h2hva 28754 | . . 3 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
8 | df-h0v 28750 | . . . 4 ⊢ 0ℎ = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
9 | 3 | fveq2i 6676 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
10 | 8, 9 | eqtr4i 2850 | . . 3 ⊢ 0ℎ = (0vec‘𝑈) |
11 | 5, 7, 10 | hladdid 28683 | . 2 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ) → (𝐴 +ℎ 0ℎ) = 𝐴) |
12 | 1, 11 | mpan 688 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 〈cop 4576 ‘cfv 6358 (class class class)co 7159 BaseSetcba 28366 0veccn0v 28368 CHilOLDchlo 28665 ℋchba 28699 +ℎ cva 28700 ·ℎ csm 28701 normℎcno 28703 0ℎc0v 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-1st 7692 df-2nd 7693 df-grpo 28273 df-gid 28274 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-nmcv 28380 df-cbn 28643 df-hlo 28666 df-hba 28749 df-h0v 28750 |
This theorem is referenced by: (None) |
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