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| Mirrors > Home > HSE Home > Th. List > axhvmul0-zf | Structured version Visualization version GIF version | ||
| Description: Derive Axiom ax-hvmul0 30941 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 30900 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 3 | axhil.1 | . . . . 5 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
| 4 | 3 | fveq2i 6819 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
| 5 | 2, 4 | eqtr4i 2755 | . . 3 ⊢ ℋ = (BaseSet‘𝑈) |
| 6 | 1 | hlnvi 30823 | . . . 4 ⊢ 𝑈 ∈ NrmCVec |
| 7 | 3, 6 | h2hsm 30906 | . . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
| 8 | df-h0v 30901 | . . . 4 ⊢ 0ℎ = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 9 | 3 | fveq2i 6819 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
| 10 | 8, 9 | eqtr4i 2755 | . . 3 ⊢ 0ℎ = (0vec‘𝑈) |
| 11 | 5, 7, 10 | hlmul0 30840 | . 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 1540 ∈ wcel 2109 〈cop 4579 ‘cfv 6476 (class class class)co 7340 0cc0 10997 BaseSetcba 30517 0veccn0v 30519 CHilOLDchlo 30816 ℋchba 30850 +ℎ cva 30851 ·ℎ csm 30852 normℎcno 30854 0ℎc0v 30855 |
| 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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-id 5508 df-po 5521 df-so 5522 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7297 df-ov 7343 df-oprab 7344 df-1st 7915 df-2nd 7916 df-er 8616 df-en 8864 df-dom 8865 df-sdom 8866 df-pnf 11139 df-mnf 11140 df-ltxr 11142 df-grpo 30424 df-gid 30425 df-ginv 30426 df-ablo 30476 df-vc 30490 df-nv 30523 df-va 30526 df-ba 30527 df-sm 30528 df-0v 30529 df-nmcv 30531 df-cbn 30794 df-hlo 30817 df-hba 30900 df-h0v 30901 |
| This theorem is referenced by: (None) |
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