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Mirrors > Home > MPE Home > Th. List > thlbas | Structured version Visualization version GIF version |
Description: Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
Ref | Expression |
---|---|
thlval.k | ⊢ 𝐾 = (toHL‘𝑊) |
thlbas.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
thlbas | ⊢ 𝐶 = (Base‘𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thlbas.c | . . . . . 6 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
2 | 1 | fvexi 6690 | . . . . 5 ⊢ 𝐶 ∈ V |
3 | eqid 2738 | . . . . . 6 ⊢ (toInc‘𝐶) = (toInc‘𝐶) | |
4 | 3 | ipobas 17883 | . . . . 5 ⊢ (𝐶 ∈ V → 𝐶 = (Base‘(toInc‘𝐶))) |
5 | 2, 4 | ax-mp 5 | . . . 4 ⊢ 𝐶 = (Base‘(toInc‘𝐶)) |
6 | baseid 16648 | . . . . 5 ⊢ Base = Slot (Base‘ndx) | |
7 | 1re 10721 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | 1nn 11729 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
9 | 1nn0 11994 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | 1lt10 12320 | . . . . . . . 8 ⊢ 1 < ;10 | |
11 | 8, 9, 9, 10 | declti 12219 | . . . . . . 7 ⊢ 1 < ;11 |
12 | 7, 11 | ltneii 10833 | . . . . . 6 ⊢ 1 ≠ ;11 |
13 | basendx 16652 | . . . . . . 7 ⊢ (Base‘ndx) = 1 | |
14 | ocndx 16778 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
15 | 13, 14 | neeq12i 3000 | . . . . . 6 ⊢ ((Base‘ndx) ≠ (oc‘ndx) ↔ 1 ≠ ;11) |
16 | 12, 15 | mpbir 234 | . . . . 5 ⊢ (Base‘ndx) ≠ (oc‘ndx) |
17 | 6, 16 | setsnid 16644 | . . . 4 ⊢ (Base‘(toInc‘𝐶)) = (Base‘((toInc‘𝐶) sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
18 | 5, 17 | eqtri 2761 | . . 3 ⊢ 𝐶 = (Base‘((toInc‘𝐶) sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
19 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
20 | eqid 2738 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
21 | 19, 1, 3, 20 | thlval 20513 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = ((toInc‘𝐶) sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
22 | 21 | fveq2d 6680 | . . 3 ⊢ (𝑊 ∈ V → (Base‘𝐾) = (Base‘((toInc‘𝐶) sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
23 | 18, 22 | eqtr4id 2792 | . 2 ⊢ (𝑊 ∈ V → 𝐶 = (Base‘𝐾)) |
24 | base0 16641 | . . 3 ⊢ ∅ = (Base‘∅) | |
25 | fvprc 6668 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (ClSubSp‘𝑊) = ∅) | |
26 | 1, 25 | syl5eq 2785 | . . 3 ⊢ (¬ 𝑊 ∈ V → 𝐶 = ∅) |
27 | fvprc 6668 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
28 | 19, 27 | syl5eq 2785 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
29 | 28 | fveq2d 6680 | . . 3 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐾) = (Base‘∅)) |
30 | 24, 26, 29 | 3eqtr4a 2799 | . 2 ⊢ (¬ 𝑊 ∈ V → 𝐶 = (Base‘𝐾)) |
31 | 23, 30 | pm2.61i 185 | 1 ⊢ 𝐶 = (Base‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ∅c0 4211 〈cop 4522 ‘cfv 6339 (class class class)co 7172 1c1 10618 ;cdc 12181 ndxcnx 16585 sSet csts 16586 Basecbs 16588 occoc 16678 toInccipo 17879 ocvcocv 20478 ClSubSpccss 20479 toHLcthl 20480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-fz 12984 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-tset 16689 df-ple 16690 df-ocomp 16691 df-ipo 17880 df-thl 20483 |
This theorem is referenced by: (None) |
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