Theorem pmaple 39763

Description: The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)

Hypotheses

Ref	Expression
pmaple.b	⊢ 𝐵 = (Base‘𝐾)
pmaple.l	⊢ ≤ = (le‘𝐾)
pmaple.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmaple	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑀‘𝑋) ⊆ (𝑀‘𝑌)))

Proof of Theorem pmaple

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlpos 39367	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
2		pmaple.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
3		eqid 2737	. . . . . . . . . 10 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4	2, 3	atbase 39290	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
5		pmaple.l	. . . . . . . . . . . . . . 15 ⊢ ≤ = (le‘𝐾)
6	2, 5	postr 18366	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Poset ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌))
7	6	exp4b 430	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Poset → ((𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
8	7	3expd 1354	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Poset → (𝑝 ∈ 𝐵 → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))))
9	8	com23 86	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Poset → (𝑋 ∈ 𝐵 → (𝑝 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))))
10	9	com34 91	. . . . . . . . . 10 ⊢ (𝐾 ∈ Poset → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))))
11	10	3imp 1111	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
12	4, 11	syl5 34	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
13	12	com34 91	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑋 ≤ 𝑌 → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))))
14	13	com23 86	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))))
15	14	ralrimdv 3152	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
16	1, 15	syl3an1 1164	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
17		ss2rab 4071	. . . 4 ⊢ ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))
18	16, 17	imbitrrdi 252	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))
19		hlclat 39359	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
20		ssrab2 4080	. . . . . . . . 9 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ (Atoms‘𝐾)
21	2, 3	atssbase 39291	. . . . . . . . 9 ⊢ (Atoms‘𝐾) ⊆ 𝐵
22	20, 21	sstri 3993	. . . . . . . 8 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵
23		eqid 2737	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
24	2, 5, 23	lubss 18558	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵 ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))
25	22, 24	mp3an2 1451	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))
26	25	ex 412	. . . . . 6 ⊢ (𝐾 ∈ CLat → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})))
27	19, 26	syl 17	. . . . 5 ⊢ (𝐾 ∈ HL → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})))
28	27	3ad2ant1 1134	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})))
29		hlomcmat 39366	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30	29	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31		simp2 1138	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
32	2, 5, 23, 3	atlatmstc 39320	. . . . . 6 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) = 𝑋)
33	30, 31, 32	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) = 𝑋)
34		simp3 1139	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
35	2, 5, 23, 3	atlatmstc 39320	. . . . . 6 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) = 𝑌)
36	30, 34, 35	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) = 𝑌)
37	33, 36	breq12d 5156	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) ↔ 𝑋 ≤ 𝑌))
38	28, 37	sylibd 239	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → 𝑋 ≤ 𝑌))
39	18, 38	impbid 212	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))
40		pmaple.m	. . . . 5 ⊢ 𝑀 = (pmap‘𝐾)
41	2, 5, 3, 40	pmapval 39759	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋})
42	41	3adant3 1133	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋})
43	2, 5, 3, 40	pmapval 39759	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})
44	43	3adant2 1132	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})
45	42, 44	sseq12d 4017	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))
46	39, 45	bitr4d 282	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑀‘𝑋) ⊆ (𝑀‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436   ⊆ wss 3951   class class class wbr 5143  ‘cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  lubclub 18355  CLatccla 18543  OMLcoml 39176  Atomscatm 39264  AtLatcal 39265  HLchlt 39351  pmapcpmap 39499

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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3062  df-rex 3071  df-rmo 3380  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-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-id 5578  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-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-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-pmap 39506

This theorem is referenced by:  pmap11  39764  hlmod1i  39858  paddunN  39929  pmapojoinN  39970  pl42N  39985