Theorem dmdi 30660

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 30657	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
2	1	biimpd 228	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 → ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)))))
3		sseq2 3952	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐶))
4		ineq1 4145	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ 𝐴) = (𝐶 ∩ 𝐴))
5	4	oveq1d 7286	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = ((𝐶 ∩ 𝐴) ∨ℋ 𝐵))
6		ineq1 4145	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)))
7	5, 6	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = 𝐶 → (((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ↔ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵))))
8	3, 7	imbi12d 345	. . . . 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 3556	. . . 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 508	. . 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 419	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 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ∀wral 3066   ∩ cin 3891   ⊆ wss 3892   class class class wbr 5079  (class class class)co 7271   C_ℋ cch 29287   ∨_ℋ chj 29291   𝑀_ℋ^* cdmd 29325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-iota 6390  df-fv 6440  df-ov 7274  df-dmd 30639

This theorem is referenced by:  dmdi2  30662  dmdsl3  30673  csmdsymi  30692  mdsymlem1  30761