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Mirrors > Home > MPE Home > Th. List > thlbasOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of thlbas 21632 as of 11-Nov-2024. Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
thlval.k | ⊢ 𝐾 = (toHL‘𝑊) |
thlbas.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
thlbasOLD | ⊢ 𝐶 = (Base‘𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thlbas.c | . . . . . 6 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
2 | 1 | fvexi 6906 | . . . . 5 ⊢ 𝐶 ∈ V |
3 | eqid 2725 | . . . . . 6 ⊢ (toInc‘𝐶) = (toInc‘𝐶) | |
4 | 3 | ipobas 18522 | . . . . 5 ⊢ (𝐶 ∈ V → 𝐶 = (Base‘(toInc‘𝐶))) |
5 | 2, 4 | ax-mp 5 | . . . 4 ⊢ 𝐶 = (Base‘(toInc‘𝐶)) |
6 | baseid 17182 | . . . . 5 ⊢ Base = Slot (Base‘ndx) | |
7 | 1re 11244 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | 1nn 12253 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
9 | 1nn0 12518 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
10 | 1lt10 12846 | . . . . . . . 8 ⊢ 1 < ;10 | |
11 | 8, 9, 9, 10 | declti 12745 | . . . . . . 7 ⊢ 1 < ;11 |
12 | 7, 11 | ltneii 11357 | . . . . . 6 ⊢ 1 ≠ ;11 |
13 | basendx 17188 | . . . . . . 7 ⊢ (Base‘ndx) = 1 | |
14 | ocndx 17361 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
15 | 13, 14 | neeq12i 2997 | . . . . . 6 ⊢ ((Base‘ndx) ≠ (oc‘ndx) ↔ 1 ≠ ;11) |
16 | 12, 15 | mpbir 230 | . . . . 5 ⊢ (Base‘ndx) ≠ (oc‘ndx) |
17 | 6, 16 | setsnid 17177 | . . . 4 ⊢ (Base‘(toInc‘𝐶)) = (Base‘((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩)) |
18 | 5, 17 | eqtri 2753 | . . 3 ⊢ 𝐶 = (Base‘((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩)) |
19 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
20 | eqid 2725 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
21 | 19, 1, 3, 20 | thlval 21631 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = ((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩)) |
22 | 21 | fveq2d 6896 | . . 3 ⊢ (𝑊 ∈ V → (Base‘𝐾) = (Base‘((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩))) |
23 | 18, 22 | eqtr4id 2784 | . 2 ⊢ (𝑊 ∈ V → 𝐶 = (Base‘𝐾)) |
24 | base0 17184 | . . 3 ⊢ ∅ = (Base‘∅) | |
25 | fvprc 6884 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (ClSubSp‘𝑊) = ∅) | |
26 | 1, 25 | eqtrid 2777 | . . 3 ⊢ (¬ 𝑊 ∈ V → 𝐶 = ∅) |
27 | fvprc 6884 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
28 | 19, 27 | eqtrid 2777 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
29 | 28 | fveq2d 6896 | . . 3 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐾) = (Base‘∅)) |
30 | 24, 26, 29 | 3eqtr4a 2791 | . 2 ⊢ (¬ 𝑊 ∈ V → 𝐶 = (Base‘𝐾)) |
31 | 23, 30 | pm2.61i 182 | 1 ⊢ 𝐶 = (Base‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ∅c0 4318 ⟨cop 4630 ‘cfv 6543 (class class class)co 7416 1c1 11139 ;cdc 12707 sSet csts 17131 ndxcnx 17161 Basecbs 17179 occoc 17240 toInccipo 18518 ocvcocv 21596 ClSubSpccss 21597 toHLcthl 21598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-tset 17251 df-ple 17252 df-ocomp 17253 df-ipo 18519 df-thl 21601 |
This theorem is referenced by: (None) |
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