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Theorem gsumbagdiaglem 19577
 Description: Lemma for gsumbagdiag 19578. (Contributed by Mario Carneiro, 5-Jan-2015.)
Hypotheses
Ref Expression
psrbag.d 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.1 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
gsumbagdiag.i (𝜑𝐼𝑉)
gsumbagdiag.f (𝜑𝐹𝐷)
Assertion
Ref Expression
gsumbagdiaglem ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)}))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑥,𝑉,𝑦   𝑓,𝐼,𝑥,𝑦   𝑥,𝑆   𝑥,𝐷,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐷(𝑓)   𝑆(𝑦,𝑓)   𝑉(𝑓)

Proof of Theorem gsumbagdiaglem
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 813 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})
2 breq1 4807 . . . . . 6 (𝑥 = 𝑌 → (𝑥𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑌𝑟 ≤ (𝐹𝑓𝑋)))
32elrab 3504 . . . . 5 (𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)} ↔ (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
41, 3sylib 208 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
54simpld 477 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝐷)
64simprd 482 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟 ≤ (𝐹𝑓𝑋))
7 gsumbagdiag.i . . . . . . 7 (𝜑𝐼𝑉)
87adantr 472 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐼𝑉)
9 gsumbagdiag.f . . . . . . 7 (𝜑𝐹𝐷)
109adantr 472 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹𝐷)
11 simprl 811 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑆)
12 breq1 4807 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑦𝑟𝐹𝑋𝑟𝐹))
13 psrbagconf1o.1 . . . . . . . . . 10 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
1412, 13elrab2 3507 . . . . . . . . 9 (𝑋𝑆 ↔ (𝑋𝐷𝑋𝑟𝐹))
1511, 14sylib 208 . . . . . . . 8 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝐷𝑋𝑟𝐹))
1615simpld 477 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝐷)
17 psrbag.d . . . . . . . 8 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
1817psrbagf 19567 . . . . . . 7 ((𝐼𝑉𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
198, 16, 18syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋:𝐼⟶ℕ0)
2015simprd 482 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟𝐹)
2117psrbagcon 19573 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝐷𝑋:𝐼⟶ℕ0𝑋𝑟𝐹)) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
228, 10, 19, 20, 21syl13anc 1479 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
2322simprd 482 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∘𝑟𝐹)
2417psrbagf 19567 . . . . . 6 ((𝐼𝑉𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
258, 5, 24syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌:𝐼⟶ℕ0)
2622simpld 477 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∈ 𝐷)
2717psrbagf 19567 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝑓𝑋) ∈ 𝐷) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
288, 26, 27syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
2917psrbagf 19567 . . . . . 6 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
308, 10, 29syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹:𝐼⟶ℕ0)
31 nn0re 11493 . . . . . . 7 (𝑢 ∈ ℕ0𝑢 ∈ ℝ)
32 nn0re 11493 . . . . . . 7 (𝑣 ∈ ℕ0𝑣 ∈ ℝ)
33 nn0re 11493 . . . . . . 7 (𝑤 ∈ ℕ0𝑤 ∈ ℝ)
34 letr 10323 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3531, 32, 33, 34syl3an 1164 . . . . . 6 ((𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3635adantl 473 . . . . 5 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ (𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0)) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
378, 25, 28, 30, 36caoftrn 7097 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝑌𝑟 ≤ (𝐹𝑓𝑋) ∧ (𝐹𝑓𝑋) ∘𝑟𝐹) → 𝑌𝑟𝐹))
386, 23, 37mp2and 717 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟𝐹)
39 breq1 4807 . . . 4 (𝑦 = 𝑌 → (𝑦𝑟𝐹𝑌𝑟𝐹))
4039, 13elrab2 3507 . . 3 (𝑌𝑆 ↔ (𝑌𝐷𝑌𝑟𝐹))
415, 38, 40sylanbrc 701 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑆)
4219ffvelrnda 6522 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
4325ffvelrnda 6522 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
4430ffvelrnda 6522 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
45 nn0re 11493 . . . . . . . 8 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℝ)
46 nn0re 11493 . . . . . . . 8 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℝ)
47 nn0re 11493 . . . . . . . 8 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℝ)
48 leaddsub2 10697 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
49 leaddsub 10696 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5048, 49bitr3d 270 . . . . . . . 8 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5145, 46, 47, 50syl3an 1164 . . . . . . 7 (((𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝐹𝑧) ∈ ℕ0) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5242, 43, 44, 51syl3anc 1477 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5352ralbidva 3123 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
54 ovexd 6843 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑋𝑧)) ∈ V)
5525feqmptd 6411 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
56 ffn 6206 . . . . . . . 8 (𝐹:𝐼⟶ℕ0𝐹 Fn 𝐼)
5730, 56syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹 Fn 𝐼)
58 ffn 6206 . . . . . . . 8 (𝑋:𝐼⟶ℕ0𝑋 Fn 𝐼)
5919, 58syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 Fn 𝐼)
60 inidm 3965 . . . . . . 7 (𝐼𝐼) = 𝐼
61 eqidd 2761 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) = (𝐹𝑧))
62 eqidd 2761 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) = (𝑋𝑧))
6357, 59, 8, 8, 60, 61, 62offval 7069 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑋𝑧))))
648, 43, 54, 55, 63ofrfval2 7080 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ ∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
65 ovexd 6843 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑌𝑧)) ∈ V)
6619feqmptd 6411 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
67 ffn 6206 . . . . . . . 8 (𝑌:𝐼⟶ℕ0𝑌 Fn 𝐼)
6825, 67syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 Fn 𝐼)
69 eqidd 2761 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) = (𝑌𝑧))
7057, 68, 8, 8, 60, 61, 69offval 7069 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑌) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑌𝑧))))
718, 42, 65, 66, 70ofrfval2 7080 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝑟 ≤ (𝐹𝑓𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
7253, 64, 713bitr4d 300 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
736, 72mpbid 222 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟 ≤ (𝐹𝑓𝑌))
74 breq1 4807 . . . 4 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝐹𝑓𝑌) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7574elrab 3504 . . 3 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7616, 73, 75sylanbrc 701 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)})
7741, 76jca 555 1 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340   class class class wbr 4804  ◡ccnv 5265   “ cima 5269   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ∘𝑓 cof 7060   ∘𝑟 cofr 7061   ↑𝑚 cmap 8023  Fincfn 8121  ℝcr 10127   + caddc 10131   ≤ cle 10267   − cmin 10458  ℕcn 11212  ℕ0cn0 11484 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485 This theorem is referenced by:  gsumbagdiag  19578  psrass1lem  19579
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