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Mirrors > Home > MPE Home > Th. List > Mathboxes > redundss3 | Structured version Visualization version GIF version |
Description: Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.) |
Ref | Expression |
---|---|
redundss3.1 | ⊢ 𝐷 ⊆ 𝐶 |
Ref | Expression |
---|---|
redundss3 | ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 → 𝐴 Redund 〈𝐵, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4174 | . . . 4 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) → ((𝐴 ∩ 𝐶) ∩ 𝐷) = ((𝐵 ∩ 𝐶) ∩ 𝐷)) | |
2 | redundss3.1 | . . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶 | |
3 | dfss 3946 | . . . . . . . 8 ⊢ (𝐷 ⊆ 𝐶 ↔ 𝐷 = (𝐷 ∩ 𝐶)) | |
4 | 2, 3 | mpbi 232 | . . . . . . 7 ⊢ 𝐷 = (𝐷 ∩ 𝐶) |
5 | incom 4171 | . . . . . . 7 ⊢ (𝐷 ∩ 𝐶) = (𝐶 ∩ 𝐷) | |
6 | 4, 5 | eqtri 2843 | . . . . . 6 ⊢ 𝐷 = (𝐶 ∩ 𝐷) |
7 | 6 | ineq2i 4179 | . . . . 5 ⊢ (𝐴 ∩ 𝐷) = (𝐴 ∩ (𝐶 ∩ 𝐷)) |
8 | inass 4189 | . . . . 5 ⊢ ((𝐴 ∩ 𝐶) ∩ 𝐷) = (𝐴 ∩ (𝐶 ∩ 𝐷)) | |
9 | 7, 8 | eqtr4i 2846 | . . . 4 ⊢ (𝐴 ∩ 𝐷) = ((𝐴 ∩ 𝐶) ∩ 𝐷) |
10 | 6 | ineq2i 4179 | . . . . 5 ⊢ (𝐵 ∩ 𝐷) = (𝐵 ∩ (𝐶 ∩ 𝐷)) |
11 | inass 4189 | . . . . 5 ⊢ ((𝐵 ∩ 𝐶) ∩ 𝐷) = (𝐵 ∩ (𝐶 ∩ 𝐷)) | |
12 | 10, 11 | eqtr4i 2846 | . . . 4 ⊢ (𝐵 ∩ 𝐷) = ((𝐵 ∩ 𝐶) ∩ 𝐷) |
13 | 1, 9, 12 | 3eqtr4g 2880 | . . 3 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) → (𝐴 ∩ 𝐷) = (𝐵 ∩ 𝐷)) |
14 | 13 | anim2i 618 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) → (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐷) = (𝐵 ∩ 𝐷))) |
15 | df-redund 35902 | . 2 ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
16 | df-redund 35902 | . 2 ⊢ (𝐴 Redund 〈𝐵, 𝐷〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐷) = (𝐵 ∩ 𝐷))) | |
17 | 14, 15, 16 | 3imtr4i 294 | 1 ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 → 𝐴 Redund 〈𝐵, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∩ cin 3928 ⊆ wss 3929 Redund wredund 35517 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-in 3936 df-ss 3945 df-redund 35902 |
This theorem is referenced by: refrelsredund2 35911 |
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