HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mddmd2 Structured version   Visualization version   GIF version

Theorem mddmd2 30090
Description: Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mddmd2 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑥C 𝐴 𝑀* 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem mddmd2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq2 5057 . . . . 5 (𝑥 = 𝑦 → (𝐴 𝑀 𝑥𝐴 𝑀 𝑦))
21cbvralvw 3435 . . . 4 (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑦C 𝐴 𝑀 𝑦)
3 mdbr 30075 . . . . . 6 ((𝐴C𝑦C ) → (𝐴 𝑀 𝑦 ↔ ∀𝑥C (𝑥𝑦 → ((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦)))))
4 incom 4163 . . . . . . . . . . . 12 ((𝐴 𝑥) ∩ 𝑦) = (𝑦 ∩ (𝐴 𝑥))
5 chjcom 29287 . . . . . . . . . . . . 13 ((𝐴C𝑥C ) → (𝐴 𝑥) = (𝑥 𝐴))
65ineq1d 4173 . . . . . . . . . . . 12 ((𝐴C𝑥C ) → ((𝐴 𝑥) ∩ 𝑦) = ((𝑥 𝐴) ∩ 𝑦))
74, 6syl5reqr 2874 . . . . . . . . . . 11 ((𝐴C𝑥C ) → ((𝑥 𝐴) ∩ 𝑦) = (𝑦 ∩ (𝐴 𝑥)))
87adantlr 714 . . . . . . . . . 10 (((𝐴C𝑦C ) ∧ 𝑥C ) → ((𝑥 𝐴) ∩ 𝑦) = (𝑦 ∩ (𝐴 𝑥)))
9 incom 4163 . . . . . . . . . . . 12 (𝐴𝑦) = (𝑦𝐴)
109oveq1i 7156 . . . . . . . . . . 11 ((𝐴𝑦) ∨ 𝑥) = ((𝑦𝐴) ∨ 𝑥)
11 chincl 29280 . . . . . . . . . . . 12 ((𝐴C𝑦C ) → (𝐴𝑦) ∈ C )
12 chjcom 29287 . . . . . . . . . . . 12 (((𝐴𝑦) ∈ C𝑥C ) → ((𝐴𝑦) ∨ 𝑥) = (𝑥 (𝐴𝑦)))
1311, 12sylan 583 . . . . . . . . . . 11 (((𝐴C𝑦C ) ∧ 𝑥C ) → ((𝐴𝑦) ∨ 𝑥) = (𝑥 (𝐴𝑦)))
1410, 13syl5reqr 2874 . . . . . . . . . 10 (((𝐴C𝑦C ) ∧ 𝑥C ) → (𝑥 (𝐴𝑦)) = ((𝑦𝐴) ∨ 𝑥))
158, 14eqeq12d 2840 . . . . . . . . 9 (((𝐴C𝑦C ) ∧ 𝑥C ) → (((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦)) ↔ (𝑦 ∩ (𝐴 𝑥)) = ((𝑦𝐴) ∨ 𝑥)))
16 eqcom 2831 . . . . . . . . 9 ((𝑦 ∩ (𝐴 𝑥)) = ((𝑦𝐴) ∨ 𝑥) ↔ ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))
1715, 16syl6bb 290 . . . . . . . 8 (((𝐴C𝑦C ) ∧ 𝑥C ) → (((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦)) ↔ ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥))))
1817imbi2d 344 . . . . . . 7 (((𝐴C𝑦C ) ∧ 𝑥C ) → ((𝑥𝑦 → ((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦))) ↔ (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
1918ralbidva 3191 . . . . . 6 ((𝐴C𝑦C ) → (∀𝑥C (𝑥𝑦 → ((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦))) ↔ ∀𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
203, 19bitrd 282 . . . . 5 ((𝐴C𝑦C ) → (𝐴 𝑀 𝑦 ↔ ∀𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
2120ralbidva 3191 . . . 4 (𝐴C → (∀𝑦C 𝐴 𝑀 𝑦 ↔ ∀𝑦C𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
222, 21syl5bb 286 . . 3 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑦C𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
23 ralcom 3346 . . 3 (∀𝑦C𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥))) ↔ ∀𝑥C𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥))))
2422, 23syl6bb 290 . 2 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑥C𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
25 dmdbr 30080 . . 3 ((𝐴C𝑥C ) → (𝐴 𝑀* 𝑥 ↔ ∀𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
2625ralbidva 3191 . 2 (𝐴C → (∀𝑥C 𝐴 𝑀* 𝑥 ↔ ∀𝑥C𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
2724, 26bitr4d 285 1 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑥C 𝐴 𝑀* 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  cin 3918  wss 3919   class class class wbr 5053  (class class class)co 7146   C cch 28710   chj 28714   𝑀 cmd 28747   𝑀* cdmd 28748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-1cn 10589  ax-addcl 10591  ax-hilex 28780  ax-hfvadd 28781  ax-hv0cl 28784  ax-hfvmul 28786
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-map 8400  df-nn 11633  df-hlim 28753  df-sh 28988  df-ch 29002  df-chj 29091  df-md 30061  df-dmd 30062
This theorem is referenced by:  atmd  30180
  Copyright terms: Public domain W3C validator