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Theorem gsumbagdiaglem 19294
Description: Lemma for gsumbagdiag 19295. (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 795 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})
2 breq1 4616 . . . . . 6 (𝑥 = 𝑌 → (𝑥𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑌𝑟 ≤ (𝐹𝑓𝑋)))
32elrab 3346 . . . . 5 (𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)} ↔ (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
41, 3sylib 208 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
54simpld 475 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝐷)
64simprd 479 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟 ≤ (𝐹𝑓𝑋))
7 gsumbagdiag.i . . . . . . 7 (𝜑𝐼𝑉)
87adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐼𝑉)
9 gsumbagdiag.f . . . . . . 7 (𝜑𝐹𝐷)
109adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹𝐷)
11 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑆)
12 breq1 4616 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑦𝑟𝐹𝑋𝑟𝐹))
13 psrbagconf1o.1 . . . . . . . . . 10 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
1412, 13elrab2 3348 . . . . . . . . 9 (𝑋𝑆 ↔ (𝑋𝐷𝑋𝑟𝐹))
1511, 14sylib 208 . . . . . . . 8 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝐷𝑋𝑟𝐹))
1615simpld 475 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝐷)
17 psrbag.d . . . . . . . 8 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
1817psrbagf 19284 . . . . . . 7 ((𝐼𝑉𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
198, 16, 18syl2anc 692 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋:𝐼⟶ℕ0)
2015simprd 479 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟𝐹)
2117psrbagcon 19290 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝐷𝑋:𝐼⟶ℕ0𝑋𝑟𝐹)) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
228, 10, 19, 20, 21syl13anc 1325 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
2322simprd 479 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∘𝑟𝐹)
2417psrbagf 19284 . . . . . 6 ((𝐼𝑉𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
258, 5, 24syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌:𝐼⟶ℕ0)
2622simpld 475 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∈ 𝐷)
2717psrbagf 19284 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝑓𝑋) ∈ 𝐷) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
288, 26, 27syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
2917psrbagf 19284 . . . . . 6 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
308, 10, 29syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹:𝐼⟶ℕ0)
31 nn0re 11245 . . . . . . 7 (𝑢 ∈ ℕ0𝑢 ∈ ℝ)
32 nn0re 11245 . . . . . . 7 (𝑣 ∈ ℕ0𝑣 ∈ ℝ)
33 nn0re 11245 . . . . . . 7 (𝑤 ∈ ℕ0𝑤 ∈ ℝ)
34 letr 10075 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3531, 32, 33, 34syl3an 1365 . . . . . 6 ((𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3635adantl 482 . . . . 5 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ (𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0)) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
378, 25, 28, 30, 36caoftrn 6885 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝑌𝑟 ≤ (𝐹𝑓𝑋) ∧ (𝐹𝑓𝑋) ∘𝑟𝐹) → 𝑌𝑟𝐹))
386, 23, 37mp2and 714 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟𝐹)
39 breq1 4616 . . . 4 (𝑦 = 𝑌 → (𝑦𝑟𝐹𝑌𝑟𝐹))
4039, 13elrab2 3348 . . 3 (𝑌𝑆 ↔ (𝑌𝐷𝑌𝑟𝐹))
415, 38, 40sylanbrc 697 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑆)
4219ffvelrnda 6315 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
4325ffvelrnda 6315 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
4430ffvelrnda 6315 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
45 nn0re 11245 . . . . . . . 8 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℝ)
46 nn0re 11245 . . . . . . . 8 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℝ)
47 nn0re 11245 . . . . . . . 8 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℝ)
48 leaddsub2 10449 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
49 leaddsub 10448 . . . . . . . . 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 1365 . . . . . . 7 (((𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝐹𝑧) ∈ ℕ0) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5242, 43, 44, 51syl3anc 1323 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5352ralbidva 2979 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
54 ovex 6632 . . . . . . 7 ((𝐹𝑧) − (𝑋𝑧)) ∈ V
5554a1i 11 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑋𝑧)) ∈ V)
5625feqmptd 6206 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
57 ffn 6002 . . . . . . . 8 (𝐹:𝐼⟶ℕ0𝐹 Fn 𝐼)
5830, 57syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹 Fn 𝐼)
59 ffn 6002 . . . . . . . 8 (𝑋:𝐼⟶ℕ0𝑋 Fn 𝐼)
6019, 59syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 Fn 𝐼)
61 inidm 3800 . . . . . . 7 (𝐼𝐼) = 𝐼
62 eqidd 2622 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) = (𝐹𝑧))
63 eqidd 2622 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) = (𝑋𝑧))
6458, 60, 8, 8, 61, 62, 63offval 6857 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑋𝑧))))
658, 43, 55, 56, 64ofrfval2 6868 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ ∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
66 ovex 6632 . . . . . . 7 ((𝐹𝑧) − (𝑌𝑧)) ∈ V
6766a1i 11 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑌𝑧)) ∈ V)
6819feqmptd 6206 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
69 ffn 6002 . . . . . . . 8 (𝑌:𝐼⟶ℕ0𝑌 Fn 𝐼)
7025, 69syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 Fn 𝐼)
71 eqidd 2622 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) = (𝑌𝑧))
7258, 70, 8, 8, 61, 62, 71offval 6857 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑌) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑌𝑧))))
738, 42, 67, 68, 72ofrfval2 6868 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝑟 ≤ (𝐹𝑓𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
7453, 65, 733bitr4d 300 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
756, 74mpbid 222 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟 ≤ (𝐹𝑓𝑌))
76 breq1 4616 . . . 4 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝐹𝑓𝑌) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7776elrab 3346 . . 3 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7816, 75, 77sylanbrc 697 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)})
7941, 78jca 554 1 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186   class class class wbr 4613  ccnv 5073  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848  𝑟 cofr 6849  𝑚 cmap 7802  Fincfn 7899  cr 9879   + caddc 9883  cle 10019  cmin 10210  cn 10964  0cn0 11236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237
This theorem is referenced by:  gsumbagdiag  19295  psrass1lem  19296
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