Theorem dmdi 32265

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 32262	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
2	1	biimpd 229	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
3		sseq2 3964	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐶))
4		ineq1 4166	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ 𝐴) = (𝐶 ∩ 𝐴))
5	4	oveq1d 7368	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))
6		ineq1 4166	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))
7	5, 6	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝐶 → (((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ↔ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵))))
8	3, 7	imbi12d 344	. . . . 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 3575	. . . 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 507	. . 3 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → (𝐵 ⊆ 𝐶 → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))))
11	10	3impa 1109	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → (𝐵 ⊆ 𝐶 → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))))
12	11	imp32 418	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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5095  (class class class)co 7353   C_ℋ cch 30892   ∨_ℋ chj 30896   𝑀_ℋ^* cdmd 30930

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-iota 6442  df-fv 6494  df-ov 7356  df-dmd 32244

This theorem is referenced by:  dmdi2  32267  dmdsl3  32278  csmdsymi  32297  mdsymlem1  32366