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Mirrors > Home > HSE Home > Th. List > dmdi | Structured version Visualization version GIF version |
Description: Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmdi | ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵))) |
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))) |
Colors of variables: wff setvar class |
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 Cℋ cch 28016 ∨ℋ chj 28020 𝑀ℋ* 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 |
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