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Mirrors > Home > MPE Home > Th. List > Mathboxes > osumcllem3N | Structured version Visualization version GIF version |
Description: Lemma for osumclN 37137. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
osumcllem.l | ⊢ ≤ = (le‘𝐾) |
osumcllem.j | ⊢ ∨ = (join‘𝐾) |
osumcllem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
osumcllem.p | ⊢ + = (+𝑃‘𝐾) |
osumcllem.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
osumcllem.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
osumcllem.m | ⊢ 𝑀 = (𝑋 + {𝑝}) |
osumcllem.u | ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) |
Ref | Expression |
---|---|
osumcllem3N | ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘𝑋) ∩ 𝑈) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4171 | . 2 ⊢ (( ⊥ ‘𝑋) ∩ 𝑈) = (𝑈 ∩ ( ⊥ ‘𝑋)) | |
2 | osumcllem.u | . . . . 5 ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) | |
3 | simp1 1131 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝐾 ∈ HL) | |
4 | simp3 1133 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑋 ⊆ ( ⊥ ‘𝑌)) | |
5 | osumcllem.a | . . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | osumcllem.c | . . . . . . . . . . . 12 ⊢ 𝐶 = (PSubCl‘𝐾) | |
7 | 5, 6 | psubclssatN 37111 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → 𝑌 ⊆ 𝐴) |
8 | 7 | 3adant3 1127 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ 𝐴) |
9 | osumcllem.o | . . . . . . . . . . 11 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
10 | 5, 9 | polssatN 37078 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘𝑌) ⊆ 𝐴) |
11 | 3, 8, 10 | syl2anc 586 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘𝑌) ⊆ 𝐴) |
12 | 4, 11 | sstrd 3970 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑋 ⊆ 𝐴) |
13 | osumcllem.p | . . . . . . . . 9 ⊢ + = (+𝑃‘𝐾) | |
14 | 5, 13, 9 | poldmj1N 37098 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘(𝑋 + 𝑌)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) |
15 | 3, 12, 8, 14 | syl3anc 1366 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘(𝑋 + 𝑌)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) |
16 | incom 4171 | . . . . . . 7 ⊢ (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋)) | |
17 | 15, 16 | syl6eq 2871 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘(𝑋 + 𝑌)) = (( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋))) |
18 | 17 | fveq2d 6667 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) = ( ⊥ ‘(( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋)))) |
19 | 2, 18 | syl5eq 2867 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑈 = ( ⊥ ‘(( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋)))) |
20 | 19 | ineq1d 4181 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑈 ∩ ( ⊥ ‘𝑋)) = (( ⊥ ‘(( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋))) ∩ ( ⊥ ‘𝑋))) |
21 | 5, 9 | polcon2N 37089 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘𝑋)) |
22 | 8, 21 | syld3an2 1406 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘𝑋)) |
23 | 5, 9 | poml5N 37124 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ ( ⊥ ‘𝑋)) → (( ⊥ ‘(( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋))) ∩ ( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌))) |
24 | 3, 12, 22, 23 | syl3anc 1366 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘(( ⊥ ‘𝑌) ∩ ( ⊥ ‘𝑋))) ∩ ( ⊥ ‘𝑋)) = ( ⊥ ‘( ⊥ ‘𝑌))) |
25 | 9, 6 | psubcli2N 37109 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
26 | 25 | 3adant3 1127 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
27 | 20, 24, 26 | 3eqtrd 2859 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑈 ∩ ( ⊥ ‘𝑋)) = 𝑌) |
28 | 1, 27 | syl5eq 2867 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘𝑋) ∩ 𝑈) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 ⊆ wss 3929 {csn 4560 ‘cfv 6348 (class class class)co 7149 lecple 16565 joincjn 17547 Atomscatm 36433 HLchlt 36520 +𝑃cpadd 36965 ⊥𝑃cpolN 37072 PSubClcpscN 37104 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-riotaBAD 36123 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-undef 7932 df-proset 17531 df-poset 17549 df-plt 17561 df-lub 17577 df-glb 17578 df-join 17579 df-meet 17580 df-p0 17642 df-p1 17643 df-lat 17649 df-clat 17711 df-oposet 36346 df-ol 36348 df-oml 36349 df-covers 36436 df-ats 36437 df-atl 36468 df-cvlat 36492 df-hlat 36521 df-psubsp 36673 df-pmap 36674 df-padd 36966 df-polarityN 37073 df-psubclN 37105 |
This theorem is referenced by: osumcllem9N 37134 |
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