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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.) (Proof shortened by AV, 11-Nov-2024.) |
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 6857 | . . . . 5 ⊢ 𝐶 ∈ V |
3 | eqid 2733 | . . . . . 6 ⊢ (toInc‘𝐶) = (toInc‘𝐶) | |
4 | 3 | ipobas 18425 | . . . . 5 ⊢ (𝐶 ∈ V → 𝐶 = (Base‘(toInc‘𝐶))) |
5 | 2, 4 | ax-mp 5 | . . . 4 ⊢ 𝐶 = (Base‘(toInc‘𝐶)) |
6 | baseid 17091 | . . . . 5 ⊢ Base = Slot (Base‘ndx) | |
7 | basendxnocndx 17269 | . . . . 5 ⊢ (Base‘ndx) ≠ (oc‘ndx) | |
8 | 6, 7 | setsnid 17086 | . . . 4 ⊢ (Base‘(toInc‘𝐶)) = (Base‘((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩)) |
9 | 5, 8 | eqtri 2761 | . . 3 ⊢ 𝐶 = (Base‘((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩)) |
10 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
11 | eqid 2733 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
12 | 10, 1, 3, 11 | thlval 21115 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = ((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩)) |
13 | 12 | fveq2d 6847 | . . 3 ⊢ (𝑊 ∈ V → (Base‘𝐾) = (Base‘((toInc‘𝐶) sSet ⟨(oc‘ndx), (ocv‘𝑊)⟩))) |
14 | 9, 13 | eqtr4id 2792 | . 2 ⊢ (𝑊 ∈ V → 𝐶 = (Base‘𝐾)) |
15 | base0 17093 | . . 3 ⊢ ∅ = (Base‘∅) | |
16 | fvprc 6835 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (ClSubSp‘𝑊) = ∅) | |
17 | 1, 16 | eqtrid 2785 | . . 3 ⊢ (¬ 𝑊 ∈ V → 𝐶 = ∅) |
18 | fvprc 6835 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
19 | 10, 18 | eqtrid 2785 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
20 | 19 | fveq2d 6847 | . . 3 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐾) = (Base‘∅)) |
21 | 15, 17, 20 | 3eqtr4a 2799 | . 2 ⊢ (¬ 𝑊 ∈ V → 𝐶 = (Base‘𝐾)) |
22 | 14, 21 | pm2.61i 182 | 1 ⊢ 𝐶 = (Base‘𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 ⟨cop 4593 ‘cfv 6497 (class class class)co 7358 sSet csts 17040 ndxcnx 17070 Basecbs 17088 occoc 17146 toInccipo 18421 ocvcocv 21080 ClSubSpccss 21081 toHLcthl 21082 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-tset 17157 df-ple 17158 df-ocomp 17159 df-ipo 18422 df-thl 21085 |
This theorem is referenced by: (None) |
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