Theorem pmaple 40017

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 39622	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
2		pmaple.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐾)
3		eqid 2736	. . . . . . . . . . 11 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4	2, 3	atbase 39545	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
5		pmaple.l	. . . . . . . . . . . . . . . 16 ⊢ ≤ = (le‘𝐾)
6	2, 5	postr 18243	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Poset ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌))
7	6	exp4b 430	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Poset → ((𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
8	7	3expd 1354	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Poset → (𝑝 ∈ 𝐵 → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))))
9	8	com23 86	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Poset → (𝑋 ∈ 𝐵 → (𝑝 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))))
10	9	com34 91	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Poset → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))))
11	10	3imp 1110	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
12	4, 11	syl5 34	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
13	1, 12	syl3an1 1163	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌))))
14	13	com34 91	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑋 ≤ 𝑌 → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))))
15	14	com23 86	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))))
16	15	imp31 417	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))
17	16	ss2rabdv 4027	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})
18	17	ex 412	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))
19		hlclat 39614	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
20		ssrab2 4032	. . . . . . . . 9 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ (Atoms‘𝐾)
21	2, 3	atssbase 39546	. . . . . . . . 9 ⊢ (Atoms‘𝐾) ⊆ 𝐵
22	20, 21	sstri 3943	. . . . . . . 8 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵
23		eqid 2736	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
24	2, 5, 23	lubss 18436	. . . . . . . 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 39621	. . . . . . 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 39575	. . . . . 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 39575	. . . . . 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 5111	. . . 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 40013	. . . 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 40013	. . . 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 3967	. 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   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3399   ⊆ wss 3901   class class class wbr 5098  ‘cfv 6492  Basecbs 17136  lecple 17184  Posetcpo 18230  lubclub 18232  CLatccla 18421  OMLcoml 39431  Atomscatm 39519  AtLatcal 39520  HLchlt 39606  pmapcpmap 39753

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-lat 18355  df-clat 18422  df-oposet 39432  df-ol 39434  df-oml 39435  df-covers 39522  df-ats 39523  df-atl 39554  df-cvlat 39578  df-hlat 39607  df-pmap 39760

This theorem is referenced by:  pmap11  40018  hlmod1i  40112  paddunN  40183  pmapojoinN  40224  pl42N  40239