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Mirrors > Home > HSE Home > Th. List > mdslmd1lem3 | Structured version Visualization version GIF version |
Description: Lemma for mdslmd1i 31313. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdslmd.1 | ⊢ 𝐴 ∈ Cℋ |
mdslmd.2 | ⊢ 𝐵 ∈ Cℋ |
mdslmd.3 | ⊢ 𝐶 ∈ Cℋ |
mdslmd.4 | ⊢ 𝐷 ∈ Cℋ |
Ref | Expression |
---|---|
mdslmd1lem3 | ⊢ ((𝑥 ∈ Cℋ ∧ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵))))) → (((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7365 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (𝑥 ∨ℋ 𝐴) = (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴)) | |
2 | 1 | sseq1d 3976 | . . . . . 6 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 ↔ (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ⊆ 𝐷)) |
3 | 1 | oveq1d 7373 | . . . . . . . 8 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) = ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶)) |
4 | 3 | ineq1d 4172 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) = (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷)) |
5 | 1 | oveq1d 7373 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷)) = ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) |
6 | 4, 5 | sseq12d 3978 | . . . . . 6 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷)) ↔ (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷)))) |
7 | 2, 6 | imbi12d 345 | . . . . 5 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) ↔ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ⊆ 𝐷 → (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))))) |
8 | sseq2 3971 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ↔ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ))) | |
9 | sseq1 3970 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (𝑥 ⊆ (𝐷 ∩ 𝐵) ↔ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ⊆ (𝐷 ∩ 𝐵))) | |
10 | 8, 9 | anbi12d 632 | . . . . . 6 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) ↔ (((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∧ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ⊆ (𝐷 ∩ 𝐵)))) |
11 | oveq1 7365 | . . . . . . . 8 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (𝑥 ∨ℋ (𝐶 ∩ 𝐵)) = (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵))) | |
12 | 11 | ineq1d 4172 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) = ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵))) |
13 | oveq1 7365 | . . . . . . 7 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))) = (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) | |
14 | 12, 13 | sseq12d 3978 | . . . . . 6 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))) ↔ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))) |
15 | 10, 14 | imbi12d 345 | . . . . 5 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → (((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) ↔ ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∧ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ⊆ (𝐷 ∩ 𝐵)) → ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))) |
16 | 7, 15 | imbi12d 345 | . . . 4 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))) ↔ (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ⊆ 𝐷 → (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∧ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ⊆ (𝐷 ∩ 𝐵)) → ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))))) |
17 | 16 | imbi2d 341 | . . 3 ⊢ (𝑥 = if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) → ((((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵)))) → (((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))) ↔ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵)))) → (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ⊆ 𝐷 → (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∧ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ⊆ (𝐷 ∩ 𝐵)) → ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))))) |
18 | mdslmd.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
19 | mdslmd.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
20 | mdslmd.3 | . . . 4 ⊢ 𝐶 ∈ Cℋ | |
21 | mdslmd.4 | . . . 4 ⊢ 𝐷 ∈ Cℋ | |
22 | h0elch 30239 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
23 | 22 | elimel 4556 | . . . 4 ⊢ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∈ Cℋ |
24 | 18, 19, 20, 21, 23 | mdslmd1lem1 31309 | . . 3 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵)))) → (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ⊆ 𝐷 → (((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∧ if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ⊆ (𝐷 ∩ 𝐵)) → ((if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (if(𝑥 ∈ Cℋ , 𝑥, 0ℋ) ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))) |
25 | 17, 24 | dedth 4545 | . 2 ⊢ (𝑥 ∈ Cℋ → (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵)))) → (((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))))) |
26 | 25 | imp 408 | 1 ⊢ ((𝑥 ∈ Cℋ ∧ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨ℋ 𝐵))))) → (((𝑥 ∨ℋ 𝐴) ⊆ 𝐷 → (((𝑥 ∨ℋ 𝐴) ∨ℋ 𝐶) ∩ 𝐷) ⊆ ((𝑥 ∨ℋ 𝐴) ∨ℋ (𝐶 ∩ 𝐷))) → ((((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐷 ∩ 𝐵)) → ((𝑥 ∨ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 ifcif 4487 class class class wbr 5106 (class class class)co 7358 Cℋ cch 29913 ∨ℋ chj 29917 0ℋc0h 29919 𝑀ℋ cmd 29950 𝑀ℋ* cdmd 29951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cc 10376 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 ax-hilex 29983 ax-hfvadd 29984 ax-hvcom 29985 ax-hvass 29986 ax-hv0cl 29987 ax-hvaddid 29988 ax-hfvmul 29989 ax-hvmulid 29990 ax-hvmulass 29991 ax-hvdistr1 29992 ax-hvdistr2 29993 ax-hvmul0 29994 ax-hfi 30063 ax-his1 30066 ax-his2 30067 ax-his3 30068 ax-his4 30069 ax-hcompl 30186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-omul 8418 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-acn 9883 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 df-sum 15577 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-pt 17331 df-prds 17334 df-xrs 17389 df-qtop 17394 df-imas 17395 df-xps 17397 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-mulg 18878 df-cntz 19102 df-cmn 19569 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-cn 22594 df-cnp 22595 df-lm 22596 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cfil 24635 df-cau 24636 df-cmet 24637 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ims 29585 df-dip 29685 df-ssp 29706 df-ph 29797 df-cbn 29847 df-hnorm 29952 df-hba 29953 df-hvsub 29955 df-hlim 29956 df-hcau 29957 df-sh 30191 df-ch 30205 df-oc 30236 df-ch0 30237 df-shs 30292 df-chj 30294 df-md 31264 df-dmd 31265 |
This theorem is referenced by: mdslmd1i 31313 |
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