Theorem pmaple 39799

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 39404	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
2		pmaple.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
3		eqid 2731	. . . . . . . . . 10 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4	2, 3	atbase 39327	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
5		pmaple.l	. . . . . . . . . . . . . . 15 ⊢ ≤ = (le‘𝐾)
6	2, 5	postr 18223	. . . . . . . . . . . . . 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 1110	. . . . . . . . 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 3130	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
16	1, 15	syl3an1 1163	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))
17		ss2rab 4021	. . . 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 39396	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
20		ssrab2 4030	. . . . . . . . 9 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ (Atoms‘𝐾)
21	2, 3	atssbase 39328	. . . . . . . . 9 ⊢ (Atoms‘𝐾) ⊆ 𝐵
22	20, 21	sstri 3944	. . . . . . . 8 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵
23		eqid 2731	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
24	2, 5, 23	lubss 18416	. . . . . . . 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 1133	. . . 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 39403	. . . . . . 7 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
30	29	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
31		simp2 1137	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
32	2, 5, 23, 3	atlatmstc 39357	. . . . . 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 1138	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
35	2, 5, 23, 3	atlatmstc 39357	. . . . . 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 5104	. . . 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 39795	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋})
42	41	3adant3 1132	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋})
43	2, 5, 3, 40	pmapval 39795	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})
44	43	3adant2 1131	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})
45	42, 44	sseq12d 3968	. 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 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3902   class class class wbr 5091  ‘cfv 6481  Basecbs 17117  lecple 17165  Posetcpo 18210  lubclub 18212  CLatccla 18401  OMLcoml 39213  Atomscatm 39301  AtLatcal 39302  HLchlt 39388  pmapcpmap 39535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-proset 18197  df-poset 18216  df-plt 18231  df-lub 18247  df-glb 18248  df-join 18249  df-meet 18250  df-p0 18326  df-lat 18335  df-clat 18402  df-oposet 39214  df-ol 39216  df-oml 39217  df-covers 39304  df-ats 39305  df-atl 39336  df-cvlat 39360  df-hlat 39389  df-pmap 39542

This theorem is referenced by:  pmap11  39800  hlmod1i  39894  paddunN  39965  pmapojoinN  40006  pl42N  40021