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Mirrors > Home > HSE Home > Th. List > axhvaddid-zf | Structured version Visualization version GIF version |
Description: Derive axiom ax-hvaddid 28708 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 28673 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | axhil.1 | . . . . 5 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
4 | 3 | fveq2i 6666 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | 2, 4 | eqtr4i 2844 | . . 3 ⊢ ℋ = (BaseSet‘𝑈) |
6 | 1 | hlnvi 28596 | . . . 4 ⊢ 𝑈 ∈ NrmCVec |
7 | 3, 6 | h2hva 28678 | . . 3 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
8 | df-h0v 28674 | . . . 4 ⊢ 0ℎ = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
9 | 3 | fveq2i 6666 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
10 | 8, 9 | eqtr4i 2844 | . . 3 ⊢ 0ℎ = (0vec‘𝑈) |
11 | 5, 7, 10 | hladdid 28607 | . 2 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ) → (𝐴 +ℎ 0ℎ) = 𝐴) |
12 | 1, 11 | mpan 686 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 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 ℋchba 28623 +ℎ cva 28624 ·ℎ csm 28625 normℎcno 28627 0ℎc0v 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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