Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgsigalem | Structured version Visualization version GIF version |
Description: Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
carsgsiga.1 | ⊢ (𝜑 → (𝑀‘∅) = 0) |
carsgsiga.2 | ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) |
Ref | Expression |
---|---|
carsgsigalem | ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → 𝑒 = 𝑓) | |
2 | 1 | uneq2d 4121 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑒 ∪ 𝑒) = (𝑒 ∪ 𝑓)) |
3 | unidm 4110 | . . . . 5 ⊢ (𝑒 ∪ 𝑒) = 𝑒 | |
4 | 2, 3 | eqtr3di 2792 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑒 ∪ 𝑓) = 𝑒) |
5 | 4 | fveq2d 6841 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘(𝑒 ∪ 𝑓)) = (𝑀‘𝑒)) |
6 | iccssxr 13275 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
7 | simp1 1136 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → 𝜑) | |
8 | carsgval.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
10 | simp2 1137 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → 𝑒 ∈ 𝒫 𝑂) | |
11 | 9, 10 | ffvelcdmd 7030 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
12 | 6, 11 | sselid 3940 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
13 | 12 | adantr 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘𝑒) ∈ ℝ*) |
14 | 1 | fveq2d 6841 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘𝑒) = (𝑀‘𝑓)) |
15 | 14, 13 | eqeltrrd 2839 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘𝑓) ∈ ℝ*) |
16 | simp3 1138 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → 𝑓 ∈ 𝒫 𝑂) | |
17 | 9, 16 | ffvelcdmd 7030 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘𝑓) ∈ (0[,]+∞)) |
18 | 17 | adantr 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘𝑓) ∈ (0[,]+∞)) |
19 | elxrge0 13302 | . . . . . 6 ⊢ ((𝑀‘𝑓) ∈ (0[,]+∞) ↔ ((𝑀‘𝑓) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝑓))) | |
20 | 19 | simprbi 497 | . . . . 5 ⊢ ((𝑀‘𝑓) ∈ (0[,]+∞) → 0 ≤ (𝑀‘𝑓)) |
21 | 18, 20 | syl 17 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → 0 ≤ (𝑀‘𝑓)) |
22 | xraddge02 31455 | . . . . 5 ⊢ (((𝑀‘𝑒) ∈ ℝ* ∧ (𝑀‘𝑓) ∈ ℝ*) → (0 ≤ (𝑀‘𝑓) → (𝑀‘𝑒) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓)))) | |
23 | 22 | imp 407 | . . . 4 ⊢ ((((𝑀‘𝑒) ∈ ℝ* ∧ (𝑀‘𝑓) ∈ ℝ*) ∧ 0 ≤ (𝑀‘𝑓)) → (𝑀‘𝑒) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
24 | 13, 15, 21, 23 | syl21anc 836 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘𝑒) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
25 | 5, 24 | eqbrtrd 5125 | . 2 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 = 𝑓) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
26 | uniprg 4880 | . . . . . . 7 ⊢ ((𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → ∪ {𝑒, 𝑓} = (𝑒 ∪ 𝑓)) | |
27 | 26 | fveq2d 6841 | . . . . . 6 ⊢ ((𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘∪ {𝑒, 𝑓}) = (𝑀‘(𝑒 ∪ 𝑓))) |
28 | 27 | 3adant1 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘∪ {𝑒, 𝑓}) = (𝑀‘(𝑒 ∪ 𝑓))) |
29 | prct 31425 | . . . . . . 7 ⊢ ((𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → {𝑒, 𝑓} ≼ ω) | |
30 | 29 | 3adant1 1130 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → {𝑒, 𝑓} ≼ ω) |
31 | prssi 4779 | . . . . . . 7 ⊢ ((𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → {𝑒, 𝑓} ⊆ 𝒫 𝑂) | |
32 | 31 | 3adant1 1130 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → {𝑒, 𝑓} ⊆ 𝒫 𝑂) |
33 | prex 5387 | . . . . . . 7 ⊢ {𝑒, 𝑓} ∈ V | |
34 | breq1 5106 | . . . . . . . . . 10 ⊢ (𝑥 = {𝑒, 𝑓} → (𝑥 ≼ ω ↔ {𝑒, 𝑓} ≼ ω)) | |
35 | sseq1 3967 | . . . . . . . . . 10 ⊢ (𝑥 = {𝑒, 𝑓} → (𝑥 ⊆ 𝒫 𝑂 ↔ {𝑒, 𝑓} ⊆ 𝒫 𝑂)) | |
36 | 34, 35 | 3anbi23d 1439 | . . . . . . . . 9 ⊢ (𝑥 = {𝑒, 𝑓} → ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) ↔ (𝜑 ∧ {𝑒, 𝑓} ≼ ω ∧ {𝑒, 𝑓} ⊆ 𝒫 𝑂))) |
37 | unieq 4874 | . . . . . . . . . . 11 ⊢ (𝑥 = {𝑒, 𝑓} → ∪ 𝑥 = ∪ {𝑒, 𝑓}) | |
38 | 37 | fveq2d 6841 | . . . . . . . . . 10 ⊢ (𝑥 = {𝑒, 𝑓} → (𝑀‘∪ 𝑥) = (𝑀‘∪ {𝑒, 𝑓})) |
39 | esumeq1 32406 | . . . . . . . . . 10 ⊢ (𝑥 = {𝑒, 𝑓} → Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) = Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦)) | |
40 | 38, 39 | breq12d 5116 | . . . . . . . . 9 ⊢ (𝑥 = {𝑒, 𝑓} → ((𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) ↔ (𝑀‘∪ {𝑒, 𝑓}) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦))) |
41 | 36, 40 | imbi12d 344 | . . . . . . . 8 ⊢ (𝑥 = {𝑒, 𝑓} → (((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) ↔ ((𝜑 ∧ {𝑒, 𝑓} ≼ ω ∧ {𝑒, 𝑓} ⊆ 𝒫 𝑂) → (𝑀‘∪ {𝑒, 𝑓}) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦)))) |
42 | carsgsiga.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) | |
43 | 41, 42 | vtoclg 3523 | . . . . . . 7 ⊢ ({𝑒, 𝑓} ∈ V → ((𝜑 ∧ {𝑒, 𝑓} ≼ ω ∧ {𝑒, 𝑓} ⊆ 𝒫 𝑂) → (𝑀‘∪ {𝑒, 𝑓}) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦))) |
44 | 33, 43 | ax-mp 5 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑒, 𝑓} ≼ ω ∧ {𝑒, 𝑓} ⊆ 𝒫 𝑂) → (𝑀‘∪ {𝑒, 𝑓}) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦)) |
45 | 7, 30, 32, 44 | syl3anc 1371 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘∪ {𝑒, 𝑓}) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦)) |
46 | 28, 45 | eqbrtrrd 5127 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦)) |
47 | 46 | adantr 481 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦)) |
48 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑦 = 𝑒) → 𝑦 = 𝑒) | |
49 | 48 | fveq2d 6841 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑦 = 𝑒) → (𝑀‘𝑦) = (𝑀‘𝑒)) |
50 | 49 | adantlr 713 | . . . 4 ⊢ ((((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) ∧ 𝑦 = 𝑒) → (𝑀‘𝑦) = (𝑀‘𝑒)) |
51 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑦 = 𝑓) → 𝑦 = 𝑓) | |
52 | 51 | fveq2d 6841 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑦 = 𝑓) → (𝑀‘𝑦) = (𝑀‘𝑓)) |
53 | 52 | adantlr 713 | . . . 4 ⊢ ((((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) ∧ 𝑦 = 𝑓) → (𝑀‘𝑦) = (𝑀‘𝑓)) |
54 | 10 | adantr 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → 𝑒 ∈ 𝒫 𝑂) |
55 | 16 | adantr 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → 𝑓 ∈ 𝒫 𝑂) |
56 | 11 | adantr 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
57 | 17 | adantr 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → (𝑀‘𝑓) ∈ (0[,]+∞)) |
58 | simpr 485 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → 𝑒 ≠ 𝑓) | |
59 | 50, 53, 54, 55, 56, 57, 58 | esumpr 32438 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → Σ*𝑦 ∈ {𝑒, 𝑓} (𝑀‘𝑦) = ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
60 | 47, 59 | breqtrd 5129 | . 2 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) ∧ 𝑒 ≠ 𝑓) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
61 | 25, 60 | pm2.61dane 3030 | 1 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 ∪ cun 3906 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 {cpr 4586 ∪ cuni 4863 class class class wbr 5103 ⟶wf 6487 ‘cfv 6491 (class class class)co 7349 ωcom 7792 ≼ cdom 8814 0cc0 10984 +∞cpnf 11119 ℝ*cxr 11121 ≤ cle 11123 +𝑒 cxad 12959 [,]cicc 13195 Σ*cesum 32399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-inf2 9510 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 ax-addf 11063 ax-mulf 11064 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-se 5586 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7607 df-om 7793 df-1st 7911 df-2nd 7912 df-supp 8060 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8581 df-map 8700 df-pm 8701 df-ixp 8769 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-fsupp 9239 df-fi 9280 df-sup 9311 df-inf 9312 df-oi 9379 df-dju 9770 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-z 12433 df-dec 12551 df-uz 12696 df-q 12802 df-rp 12844 df-xneg 12961 df-xadd 12962 df-xmul 12963 df-ioo 13196 df-ioc 13197 df-ico 13198 df-icc 13199 df-fz 13353 df-fzo 13496 df-fl 13625 df-mod 13703 df-seq 13835 df-exp 13896 df-fac 14101 df-bc 14130 df-hash 14158 df-shft 14885 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-limsup 15287 df-clim 15304 df-rlim 15305 df-sum 15505 df-ef 15884 df-sin 15886 df-cos 15887 df-pi 15889 df-struct 16953 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-starv 17082 df-sca 17083 df-vsca 17084 df-ip 17085 df-tset 17086 df-ple 17087 df-ds 17089 df-unif 17090 df-hom 17091 df-cco 17092 df-rest 17238 df-topn 17239 df-0g 17257 df-gsum 17258 df-topgen 17259 df-pt 17260 df-prds 17263 df-ordt 17317 df-xrs 17318 df-qtop 17323 df-imas 17324 df-xps 17326 df-mre 17400 df-mrc 17401 df-acs 17403 df-ps 18389 df-tsr 18390 df-plusf 18430 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-mhm 18535 df-submnd 18536 df-grp 18685 df-minusg 18686 df-sbg 18687 df-mulg 18806 df-subg 18857 df-cntz 19029 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-cring 19891 df-subrg 20143 df-abv 20199 df-lmod 20247 df-scaf 20248 df-sra 20556 df-rgmod 20557 df-psmet 20711 df-xmet 20712 df-met 20713 df-bl 20714 df-mopn 20715 df-fbas 20716 df-fg 20717 df-cnfld 20720 df-top 22165 df-topon 22182 df-topsp 22204 df-bases 22218 df-cld 22292 df-ntr 22293 df-cls 22294 df-nei 22371 df-lp 22409 df-perf 22410 df-cn 22500 df-cnp 22501 df-haus 22588 df-tx 22835 df-hmeo 23028 df-fil 23119 df-fm 23211 df-flim 23212 df-flf 23213 df-tmd 23345 df-tgp 23346 df-tsms 23400 df-trg 23433 df-xms 23595 df-ms 23596 df-tms 23597 df-nm 23860 df-ngp 23861 df-nrg 23863 df-nlm 23864 df-ii 24162 df-cncf 24163 df-limc 25152 df-dv 25153 df-log 25834 df-esum 32400 |
This theorem is referenced by: fiunelcarsg 32689 carsgclctunlem3 32693 |
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