Theorem pmaple 39880

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 39485	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
2		pmaple.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐾)
3		eqid 2733	. . . . . . . . . . 11 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4	2, 3	atbase 39408	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
5		pmaple.l	. . . . . . . . . . . . . . . 16 ⊢ ≤ = (le‘𝐾)
6	2, 5	postr 18228	. . . . . . . . . . . . . . 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 4024	. . . 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 39477	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
20		ssrab2 4029	. . . . . . . . 9 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ (Atoms‘𝐾)
21	2, 3	atssbase 39409	. . . . . . . . 9 ⊢ (Atoms‘𝐾) ⊆ 𝐵
22	20, 21	sstri 3940	. . . . . . . 8 ⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵
23		eqid 2733	. . . . . . . . 9 ⊢ (lub‘𝐾) = (lub‘𝐾)
24	2, 5, 23	lubss 18421	. . . . . . . 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 39484	. . . . . . 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 39438	. . . . . 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 39438	. . . . . 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 5106	. . . 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 39876	. . . 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 39876	. . . 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 3964	. 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 3396   ⊆ wss 3898   class class class wbr 5093  ‘cfv 6486  Basecbs 17122  lecple 17170  Posetcpo 18215  lubclub 18217  CLatccla 18406  OMLcoml 39294  Atomscatm 39382  AtLatcal 39383  HLchlt 39469  pmapcpmap 39616

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-proset 18202  df-poset 18221  df-plt 18236  df-lub 18252  df-glb 18253  df-join 18254  df-meet 18255  df-p0 18331  df-lat 18340  df-clat 18407  df-oposet 39295  df-ol 39297  df-oml 39298  df-covers 39385  df-ats 39386  df-atl 39417  df-cvlat 39441  df-hlat 39470  df-pmap 39623

This theorem is referenced by:  pmap11  39881  hlmod1i  39975  paddunN  40046  pmapojoinN  40087  pl42N  40102