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Mirrors > Home > HSE Home > Th. List > axhvmul0-zf | Structured version Visualization version GIF version |
Description: Derive Axiom ax-hvmul0 28877 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 |
---|---|
axhvmul0-zf | ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axhil.2 | . 2 ⊢ 𝑈 ∈ CHilOLD | |
2 | df-hba 28836 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | axhil.1 | . . . . 5 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
4 | 3 | fveq2i 6654 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | 2, 4 | eqtr4i 2785 | . . 3 ⊢ ℋ = (BaseSet‘𝑈) |
6 | 1 | hlnvi 28759 | . . . 4 ⊢ 𝑈 ∈ NrmCVec |
7 | 3, 6 | h2hsm 28842 | . . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
8 | df-h0v 28837 | . . . 4 ⊢ 0ℎ = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
9 | 3 | fveq2i 6654 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
10 | 8, 9 | eqtr4i 2785 | . . 3 ⊢ 0ℎ = (0vec‘𝑈) |
11 | 5, 7, 10 | hlmul0 28776 | . 2 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ ℋ) → (0 ·ℎ 𝐴) = 0ℎ) |
12 | 1, 11 | mpan 690 | 1 ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 〈cop 4521 ‘cfv 6328 (class class class)co 7143 0cc0 10560 BaseSetcba 28453 0veccn0v 28455 CHilOLDchlo 28752 ℋchba 28786 +ℎ cva 28787 ·ℎ csm 28788 normℎcno 28790 0ℎc0v 28791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-1st 7686 df-2nd 7687 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-ltxr 10703 df-grpo 28360 df-gid 28361 df-ginv 28362 df-ablo 28412 df-vc 28426 df-nv 28459 df-va 28462 df-ba 28463 df-sm 28464 df-0v 28465 df-nmcv 28467 df-cbn 28730 df-hlo 28753 df-hba 28836 df-h0v 28837 |
This theorem is referenced by: (None) |
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