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Theorem pmapglbx 38943
Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 38944, where we read 𝑆 as 𝑆(𝑖). Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
Hypotheses
Ref Expression
pmapglb.b 𝐡 = (Baseβ€˜πΎ)
pmapglb.g 𝐺 = (glbβ€˜πΎ)
pmapglb.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
pmapglbx ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ (π‘€β€˜(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (π‘€β€˜π‘†))
Distinct variable groups:   𝑦,𝑖,𝐡   𝑖,𝐼,𝑦   𝑖,𝐾,𝑦   𝑦,𝑆
Allowed substitution hints:   𝑆(𝑖)   𝐺(𝑦,𝑖)   𝑀(𝑦,𝑖)

Proof of Theorem pmapglbx
Dummy variables 𝑝 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlclat 38531 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
21ad2antrr 722 . . . . . . 7 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ CLat)
3 pmapglb.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
4 eqid 2730 . . . . . . . . 9 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
53, 4atbase 38462 . . . . . . . 8 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ 𝐡)
65adantl 480 . . . . . . 7 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ 𝐡)
7 r19.29 3112 . . . . . . . . . . 11 ((βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆) β†’ βˆƒπ‘– ∈ 𝐼 (𝑆 ∈ 𝐡 ∧ 𝑦 = 𝑆))
8 eleq1a 2826 . . . . . . . . . . . . 13 (𝑆 ∈ 𝐡 β†’ (𝑦 = 𝑆 β†’ 𝑦 ∈ 𝐡))
98imp 405 . . . . . . . . . . . 12 ((𝑆 ∈ 𝐡 ∧ 𝑦 = 𝑆) β†’ 𝑦 ∈ 𝐡)
109rexlimivw 3149 . . . . . . . . . . 11 (βˆƒπ‘– ∈ 𝐼 (𝑆 ∈ 𝐡 ∧ 𝑦 = 𝑆) β†’ 𝑦 ∈ 𝐡)
117, 10syl 17 . . . . . . . . . 10 ((βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆) β†’ 𝑦 ∈ 𝐡)
1211ex 411 . . . . . . . . 9 (βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 β†’ (βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆 β†’ 𝑦 ∈ 𝐡))
1312ad2antlr 723 . . . . . . . 8 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆 β†’ 𝑦 ∈ 𝐡))
1413abssdv 4064 . . . . . . 7 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} βŠ† 𝐡)
15 eqid 2730 . . . . . . . 8 (leβ€˜πΎ) = (leβ€˜πΎ)
16 pmapglb.g . . . . . . . 8 𝐺 = (glbβ€˜πΎ)
173, 15, 16clatleglb 18475 . . . . . . 7 ((𝐾 ∈ CLat ∧ 𝑝 ∈ 𝐡 ∧ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} βŠ† 𝐡) β†’ (𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}𝑝(leβ€˜πΎ)𝑧))
182, 6, 14, 17syl3anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}𝑝(leβ€˜πΎ)𝑧))
19 vex 3476 . . . . . . . . . . . . 13 𝑧 ∈ V
20 eqeq1 2734 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 β†’ (𝑦 = 𝑆 ↔ 𝑧 = 𝑆))
2120rexbidv 3176 . . . . . . . . . . . . 13 (𝑦 = 𝑧 β†’ (βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆 ↔ βˆƒπ‘– ∈ 𝐼 𝑧 = 𝑆))
2219, 21elab 3667 . . . . . . . . . . . 12 (𝑧 ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} ↔ βˆƒπ‘– ∈ 𝐼 𝑧 = 𝑆)
2322imbi1i 348 . . . . . . . . . . 11 ((𝑧 ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ (βˆƒπ‘– ∈ 𝐼 𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧))
24 r19.23v 3180 . . . . . . . . . . 11 (βˆ€π‘– ∈ 𝐼 (𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ (βˆƒπ‘– ∈ 𝐼 𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧))
2523, 24bitr4i 277 . . . . . . . . . 10 ((𝑧 ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ βˆ€π‘– ∈ 𝐼 (𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧))
2625albii 1819 . . . . . . . . 9 (βˆ€π‘§(𝑧 ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ βˆ€π‘§βˆ€π‘– ∈ 𝐼 (𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧))
27 df-ral 3060 . . . . . . . . 9 (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}𝑝(leβ€˜πΎ)𝑧 ↔ βˆ€π‘§(𝑧 ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} β†’ 𝑝(leβ€˜πΎ)𝑧))
28 ralcom4 3281 . . . . . . . . 9 (βˆ€π‘– ∈ 𝐼 βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ βˆ€π‘§βˆ€π‘– ∈ 𝐼 (𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧))
2926, 27, 283bitr4i 302 . . . . . . . 8 (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}𝑝(leβ€˜πΎ)𝑧 ↔ βˆ€π‘– ∈ 𝐼 βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧))
30 nfv 1915 . . . . . . . . . . 11 Ⅎ𝑧 𝑝(leβ€˜πΎ)𝑆
31 breq2 5151 . . . . . . . . . . 11 (𝑧 = 𝑆 β†’ (𝑝(leβ€˜πΎ)𝑧 ↔ 𝑝(leβ€˜πΎ)𝑆))
3230, 31ceqsalg 3506 . . . . . . . . . 10 (𝑆 ∈ 𝐡 β†’ (βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ 𝑝(leβ€˜πΎ)𝑆))
3332ralimi 3081 . . . . . . . . 9 (βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 β†’ βˆ€π‘– ∈ 𝐼 (βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ 𝑝(leβ€˜πΎ)𝑆))
34 ralbi 3101 . . . . . . . . 9 (βˆ€π‘– ∈ 𝐼 (βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ 𝑝(leβ€˜πΎ)𝑆) β†’ (βˆ€π‘– ∈ 𝐼 βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆))
3533, 34syl 17 . . . . . . . 8 (βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 β†’ (βˆ€π‘– ∈ 𝐼 βˆ€π‘§(𝑧 = 𝑆 β†’ 𝑝(leβ€˜πΎ)𝑧) ↔ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆))
3629, 35bitrid 282 . . . . . . 7 (βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 β†’ (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}𝑝(leβ€˜πΎ)𝑧 ↔ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆))
3736ad2antlr 723 . . . . . 6 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}𝑝(leβ€˜πΎ)𝑧 ↔ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆))
3818, 37bitrd 278 . . . . 5 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ↔ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆))
3938rabbidva 3437 . . . 4 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) β†’ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})} = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆})
40393adant3 1130 . . 3 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})} = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆})
41 simp1 1134 . . . 4 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ 𝐾 ∈ HL)
4212abssdv 4064 . . . . . 6 (βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 β†’ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} βŠ† 𝐡)
433, 16clatglbcl 18462 . . . . . 6 ((𝐾 ∈ CLat ∧ {𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆} βŠ† 𝐡) β†’ (πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐡)
441, 42, 43syl2an 594 . . . . 5 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) β†’ (πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐡)
45443adant3 1130 . . . 4 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ (πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐡)
46 pmapglb.m . . . . 5 𝑀 = (pmapβ€˜πΎ)
473, 15, 4, 46pmapval 38931 . . . 4 ((𝐾 ∈ HL ∧ (πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐡) β†’ (π‘€β€˜(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})})
4841, 45, 47syl2anc 582 . . 3 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ (π‘€β€˜(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})})
49 iinrab 5071 . . . 4 (𝐼 β‰  βˆ… β†’ ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)𝑆} = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆})
50493ad2ant3 1133 . . 3 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)𝑆} = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ βˆ€π‘– ∈ 𝐼 𝑝(leβ€˜πΎ)𝑆})
5140, 48, 503eqtr4d 2780 . 2 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ (π‘€β€˜(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)𝑆})
52 nfv 1915 . . . 4 Ⅎ𝑖 𝐾 ∈ HL
53 nfra1 3279 . . . 4 β„²π‘–βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡
54 nfv 1915 . . . 4 Ⅎ𝑖 𝐼 β‰  βˆ…
5552, 53, 54nf3an 1902 . . 3 Ⅎ𝑖(𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…)
56 simpl1 1189 . . . 4 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) ∧ 𝑖 ∈ 𝐼) β†’ 𝐾 ∈ HL)
57 rspa 3243 . . . . 5 ((βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝑖 ∈ 𝐼) β†’ 𝑆 ∈ 𝐡)
58573ad2antl2 1184 . . . 4 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) ∧ 𝑖 ∈ 𝐼) β†’ 𝑆 ∈ 𝐡)
593, 15, 4, 46pmapval 38931 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐡) β†’ (π‘€β€˜π‘†) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)𝑆})
6056, 58, 59syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) ∧ 𝑖 ∈ 𝐼) β†’ (π‘€β€˜π‘†) = {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)𝑆})
6155, 60iineq2d 5019 . 2 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ ∩ 𝑖 ∈ 𝐼 (π‘€β€˜π‘†) = ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atomsβ€˜πΎ) ∣ 𝑝(leβ€˜πΎ)𝑆})
6251, 61eqtr4d 2773 1 ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡 ∧ 𝐼 β‰  βˆ…) β†’ (π‘€β€˜(πΊβ€˜{𝑦 ∣ βˆƒπ‘– ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (π‘€β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085  βˆ€wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   βŠ† wss 3947  βˆ…c0 4321  βˆ© ciin 4997   class class class wbr 5147  β€˜cfv 6542  Basecbs 17148  lecple 17208  glbcglb 18267  CLatccla 18455  Atomscatm 38436  HLchlt 38523  pmapcpmap 38671
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-poset 18270  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-lat 18389  df-clat 18456  df-ats 38440  df-hlat 38524  df-pmap 38678
This theorem is referenced by:  pmapglb  38944  pmapglb2xN  38946
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