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| Mirrors > Home > MPE Home > Th. List > thlleOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of thlle 21716 as of 11-Nov-2024. Ordering on 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) |
| thlle.i | ⊢ 𝐼 = (toInc‘𝐶) |
| thlle.l | ⊢ ≤ = (le‘𝐼) |
| Ref | Expression |
|---|---|
| thlleOLD | ⊢ ≤ = (le‘𝐾) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | thlle.l | . . . 4 ⊢ ≤ = (le‘𝐼) | |
| 2 | pleid 17411 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
| 3 | 10re 12752 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
| 4 | 1nn0 12542 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
| 5 | 0nn0 12541 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
| 6 | 1nn 12277 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
| 7 | 0lt1 11785 | . . . . . . . 8 ⊢ 0 < 1 | |
| 8 | 4, 5, 6, 7 | declt 12761 | . . . . . . 7 ⊢ ;10 < ;11 |
| 9 | 3, 8 | ltneii 11374 | . . . . . 6 ⊢ ;10 ≠ ;11 |
| 10 | plendx 17410 | . . . . . . 7 ⊢ (le‘ndx) = ;10 | |
| 11 | ocndx 17425 | . . . . . . 7 ⊢ (oc‘ndx) = ;11 | |
| 12 | 10, 11 | neeq12i 3007 | . . . . . 6 ⊢ ((le‘ndx) ≠ (oc‘ndx) ↔ ;10 ≠ ;11) |
| 13 | 9, 12 | mpbir 231 | . . . . 5 ⊢ (le‘ndx) ≠ (oc‘ndx) |
| 14 | 2, 13 | setsnid 17245 | . . . 4 ⊢ (le‘𝐼) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
| 15 | 1, 14 | eqtri 2765 | . . 3 ⊢ ≤ = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
| 16 | thlval.k | . . . . 5 ⊢ 𝐾 = (toHL‘𝑊) | |
| 17 | thlbas.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
| 18 | thlle.i | . . . . 5 ⊢ 𝐼 = (toInc‘𝐶) | |
| 19 | eqid 2737 | . . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
| 20 | 16, 17, 18, 19 | thlval 21713 | . . . 4 ⊢ (𝑊 ∈ V → 𝐾 = (𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉)) |
| 21 | 20 | fveq2d 6910 | . . 3 ⊢ (𝑊 ∈ V → (le‘𝐾) = (le‘(𝐼 sSet 〈(oc‘ndx), (ocv‘𝑊)〉))) |
| 22 | 15, 21 | eqtr4id 2796 | . 2 ⊢ (𝑊 ∈ V → ≤ = (le‘𝐾)) |
| 23 | 2 | str0 17226 | . . 3 ⊢ ∅ = (le‘∅) |
| 24 | 17 | fvexi 6920 | . . . . . 6 ⊢ 𝐶 ∈ V |
| 25 | 18 | ipolerval 18577 | . . . . . 6 ⊢ (𝐶 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |
| 26 | 24, 25 | ax-mp 5 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼) |
| 27 | 1, 26 | eqtr4i 2768 | . . . 4 ⊢ ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} |
| 28 | opabn0 5558 | . . . . . 6 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)) | |
| 29 | vex 3484 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
| 30 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 31 | 29, 30 | prss 4820 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ↔ {𝑥, 𝑦} ⊆ 𝐶) |
| 32 | elfvex 6944 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ V) | |
| 33 | 32, 17 | eleq2s 2859 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑊 ∈ V) |
| 34 | 33 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
| 35 | 31, 34 | sylanbr 582 | . . . . . . 7 ⊢ (({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
| 36 | 35 | exlimivv 1932 | . . . . . 6 ⊢ (∃𝑥∃𝑦({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦) → 𝑊 ∈ V) |
| 37 | 28, 36 | sylbi 217 | . . . . 5 ⊢ ({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} ≠ ∅ → 𝑊 ∈ V) |
| 38 | 37 | necon1bi 2969 | . . . 4 ⊢ (¬ 𝑊 ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐶 ∧ 𝑥 ⊆ 𝑦)} = ∅) |
| 39 | 27, 38 | eqtrid 2789 | . . 3 ⊢ (¬ 𝑊 ∈ V → ≤ = ∅) |
| 40 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (toHL‘𝑊) = ∅) | |
| 41 | 16, 40 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
| 42 | 41 | fveq2d 6910 | . . 3 ⊢ (¬ 𝑊 ∈ V → (le‘𝐾) = (le‘∅)) |
| 43 | 23, 39, 42 | 3eqtr4a 2803 | . 2 ⊢ (¬ 𝑊 ∈ V → ≤ = (le‘𝐾)) |
| 44 | 22, 43 | pm2.61i 182 | 1 ⊢ ≤ = (le‘𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 {cpr 4628 〈cop 4632 {copab 5205 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ;cdc 12733 sSet csts 17200 ndxcnx 17230 lecple 17304 occoc 17305 toInccipo 18572 ocvcocv 21678 ClSubSpccss 21679 toHLcthl 21680 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-tset 17316 df-ple 17317 df-ocomp 17318 df-ipo 18573 df-thl 21683 |
| This theorem is referenced by: (None) |
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