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Theorem mddmd2 29723
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 4877 . . . . 5 (𝑥 = 𝑦 → (𝐴 𝑀 𝑥𝐴 𝑀 𝑦))
21cbvralv 3383 . . . 4 (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑦C 𝐴 𝑀 𝑦)
3 mdbr 29708 . . . . . 6 ((𝐴C𝑦C ) → (𝐴 𝑀 𝑦 ↔ ∀𝑥C (𝑥𝑦 → ((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦)))))
4 incom 4032 . . . . . . . . . . . 12 ((𝐴 𝑥) ∩ 𝑦) = (𝑦 ∩ (𝐴 𝑥))
5 chjcom 28920 . . . . . . . . . . . . 13 ((𝐴C𝑥C ) → (𝐴 𝑥) = (𝑥 𝐴))
65ineq1d 4040 . . . . . . . . . . . 12 ((𝐴C𝑥C ) → ((𝐴 𝑥) ∩ 𝑦) = ((𝑥 𝐴) ∩ 𝑦))
74, 6syl5reqr 2876 . . . . . . . . . . 11 ((𝐴C𝑥C ) → ((𝑥 𝐴) ∩ 𝑦) = (𝑦 ∩ (𝐴 𝑥)))
87adantlr 708 . . . . . . . . . 10 (((𝐴C𝑦C ) ∧ 𝑥C ) → ((𝑥 𝐴) ∩ 𝑦) = (𝑦 ∩ (𝐴 𝑥)))
9 incom 4032 . . . . . . . . . . . 12 (𝐴𝑦) = (𝑦𝐴)
109oveq1i 6915 . . . . . . . . . . 11 ((𝐴𝑦) ∨ 𝑥) = ((𝑦𝐴) ∨ 𝑥)
11 chincl 28913 . . . . . . . . . . . 12 ((𝐴C𝑦C ) → (𝐴𝑦) ∈ C )
12 chjcom 28920 . . . . . . . . . . . 12 (((𝐴𝑦) ∈ C𝑥C ) → ((𝐴𝑦) ∨ 𝑥) = (𝑥 (𝐴𝑦)))
1311, 12sylan 577 . . . . . . . . . . 11 (((𝐴C𝑦C ) ∧ 𝑥C ) → ((𝐴𝑦) ∨ 𝑥) = (𝑥 (𝐴𝑦)))
1410, 13syl5reqr 2876 . . . . . . . . . 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 2832 . . . . . . . . 9 ((𝑦 ∩ (𝐴 𝑥)) = ((𝑦𝐴) ∨ 𝑥) ↔ ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))
1715, 16syl6bb 279 . . . . . . . 8 (((𝐴C𝑦C ) ∧ 𝑥C ) → (((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦)) ↔ ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥))))
1817imbi2d 332 . . . . . . 7 (((𝐴C𝑦C ) ∧ 𝑥C ) → ((𝑥𝑦 → ((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦))) ↔ (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
1918ralbidva 3194 . . . . . 6 ((𝐴C𝑦C ) → (∀𝑥C (𝑥𝑦 → ((𝑥 𝐴) ∩ 𝑦) = (𝑥 (𝐴𝑦))) ↔ ∀𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
203, 19bitrd 271 . . . . 5 ((𝐴C𝑦C ) → (𝐴 𝑀 𝑦 ↔ ∀𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
2120ralbidva 3194 . . . 4 (𝐴C → (∀𝑦C 𝐴 𝑀 𝑦 ↔ ∀𝑦C𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
222, 21syl5bb 275 . . 3 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑦C𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
23 ralcom 3308 . . 3 (∀𝑦C𝑥C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥))) ↔ ∀𝑥C𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥))))
2422, 23syl6bb 279 . 2 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑥C𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
25 dmdbr 29713 . . 3 ((𝐴C𝑥C ) → (𝐴 𝑀* 𝑥 ↔ ∀𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
2625ralbidva 3194 . 2 (𝐴C → (∀𝑥C 𝐴 𝑀* 𝑥 ↔ ∀𝑥C𝑦C (𝑥𝑦 → ((𝑦𝐴) ∨ 𝑥) = (𝑦 ∩ (𝐴 𝑥)))))
2724, 26bitr4d 274 1 (𝐴C → (∀𝑥C 𝐴 𝑀 𝑥 ↔ ∀𝑥C 𝐴 𝑀* 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3117  cin 3797  wss 3798   class class class wbr 4873  (class class class)co 6905   C cch 28341   chj 28345   𝑀 cmd 28378   𝑀* cdmd 28379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-1cn 10310  ax-addcl 10312  ax-hilex 28411  ax-hfvadd 28412  ax-hv0cl 28415  ax-hfvmul 28417
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-map 8124  df-nn 11351  df-hlim 28384  df-sh 28619  df-ch 28633  df-chj 28724  df-md 29694  df-dmd 29695
This theorem is referenced by:  atmd  29813
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