Theorem dmdi 29391

Description: Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
dmdi	⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))

Proof of Theorem dmdi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmdbr 29388	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
2	1	biimpd 219	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
3		sseq2 3733	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐶))
4		ineq1 3915	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ 𝐴) = (𝐶 ∩ 𝐴))
5	4	oveq1d 6780	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))
6		ineq1 3915	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))
7	5, 6	eqeq12d 2739	. . . . . 6 ⊢ (𝑥 = 𝐶 → (((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ↔ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵))))
8	3, 7	imbi12d 333	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))) ↔ (𝐵 ⊆ 𝐶 → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))))
9	8	rspcv 3409	. . . 4 ⊢ (𝐶 ∈ Cℋ → (∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))) → (𝐵 ⊆ 𝐶 → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))))
10	2, 9	sylan9 692	. . 3 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → (𝐵 ⊆ 𝐶 → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))))
11	10	3impa 1100	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → (𝐵 ⊆ 𝐶 → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))))
12	11	imp32 448	1 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103  ∀wral 3014   ∩ cin 3679   ⊆ wss 3680   class class class wbr 4760  (class class class)co 6765   C_ℋ cch 28016   ∨_ℋ chj 28020   𝑀_ℋ^* cdmd 28054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-iota 5964  df-fv 6009  df-ov 6768  df-dmd 29370

This theorem is referenced by:  dmdi2  29393  dmdsl3  29404  csmdsymi  29423  mdsymlem1  29492