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Theorem gsumbagdiaglem 19196
 Description: Lemma for gsumbagdiag 19197. (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 792 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})
2 breq1 4586 . . . . . 6 (𝑥 = 𝑌 → (𝑥𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑌𝑟 ≤ (𝐹𝑓𝑋)))
32elrab 3331 . . . . 5 (𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)} ↔ (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
41, 3sylib 207 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
54simpld 474 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝐷)
64simprd 478 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟 ≤ (𝐹𝑓𝑋))
7 gsumbagdiag.i . . . . . . 7 (𝜑𝐼𝑉)
87adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐼𝑉)
9 gsumbagdiag.f . . . . . . 7 (𝜑𝐹𝐷)
109adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹𝐷)
11 simprl 790 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑆)
12 breq1 4586 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑦𝑟𝐹𝑋𝑟𝐹))
13 psrbagconf1o.1 . . . . . . . . . 10 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
1412, 13elrab2 3333 . . . . . . . . 9 (𝑋𝑆 ↔ (𝑋𝐷𝑋𝑟𝐹))
1511, 14sylib 207 . . . . . . . 8 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝐷𝑋𝑟𝐹))
1615simpld 474 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝐷)
17 psrbag.d . . . . . . . 8 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
1817psrbagf 19186 . . . . . . 7 ((𝐼𝑉𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
198, 16, 18syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋:𝐼⟶ℕ0)
2015simprd 478 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟𝐹)
2117psrbagcon 19192 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝐷𝑋:𝐼⟶ℕ0𝑋𝑟𝐹)) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
228, 10, 19, 20, 21syl13anc 1320 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
2322simprd 478 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∘𝑟𝐹)
2417psrbagf 19186 . . . . . 6 ((𝐼𝑉𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
258, 5, 24syl2anc 691 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌:𝐼⟶ℕ0)
2622simpld 474 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∈ 𝐷)
2717psrbagf 19186 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝑓𝑋) ∈ 𝐷) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
288, 26, 27syl2anc 691 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
2917psrbagf 19186 . . . . . 6 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
308, 10, 29syl2anc 691 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹:𝐼⟶ℕ0)
31 nn0re 11178 . . . . . . 7 (𝑢 ∈ ℕ0𝑢 ∈ ℝ)
32 nn0re 11178 . . . . . . 7 (𝑣 ∈ ℕ0𝑣 ∈ ℝ)
33 nn0re 11178 . . . . . . 7 (𝑤 ∈ ℕ0𝑤 ∈ ℝ)
34 letr 10010 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3531, 32, 33, 34syl3an 1360 . . . . . 6 ((𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3635adantl 481 . . . . 5 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ (𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0)) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
378, 25, 28, 30, 36caoftrn 6830 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝑌𝑟 ≤ (𝐹𝑓𝑋) ∧ (𝐹𝑓𝑋) ∘𝑟𝐹) → 𝑌𝑟𝐹))
386, 23, 37mp2and 711 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟𝐹)
39 breq1 4586 . . . 4 (𝑦 = 𝑌 → (𝑦𝑟𝐹𝑌𝑟𝐹))
4039, 13elrab2 3333 . . 3 (𝑌𝑆 ↔ (𝑌𝐷𝑌𝑟𝐹))
415, 38, 40sylanbrc 695 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑆)
4219ffvelrnda 6267 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
4325ffvelrnda 6267 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
4430ffvelrnda 6267 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
45 nn0re 11178 . . . . . . . 8 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℝ)
46 nn0re 11178 . . . . . . . 8 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℝ)
47 nn0re 11178 . . . . . . . 8 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℝ)
48 leaddsub2 10384 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
49 leaddsub 10383 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5048, 49bitr3d 269 . . . . . . . 8 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5145, 46, 47, 50syl3an 1360 . . . . . . 7 (((𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝐹𝑧) ∈ ℕ0) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5242, 43, 44, 51syl3anc 1318 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5352ralbidva 2968 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
54 ovex 6577 . . . . . . 7 ((𝐹𝑧) − (𝑋𝑧)) ∈ V
5554a1i 11 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑋𝑧)) ∈ V)
5625feqmptd 6159 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
57 ffn 5958 . . . . . . . 8 (𝐹:𝐼⟶ℕ0𝐹 Fn 𝐼)
5830, 57syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹 Fn 𝐼)
59 ffn 5958 . . . . . . . 8 (𝑋:𝐼⟶ℕ0𝑋 Fn 𝐼)
6019, 59syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 Fn 𝐼)
61 inidm 3784 . . . . . . 7 (𝐼𝐼) = 𝐼
62 eqidd 2611 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) = (𝐹𝑧))
63 eqidd 2611 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) = (𝑋𝑧))
6458, 60, 8, 8, 61, 62, 63offval 6802 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑋𝑧))))
658, 43, 55, 56, 64ofrfval2 6813 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ ∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
66 ovex 6577 . . . . . . 7 ((𝐹𝑧) − (𝑌𝑧)) ∈ V
6766a1i 11 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑌𝑧)) ∈ V)
6819feqmptd 6159 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
69 ffn 5958 . . . . . . . 8 (𝑌:𝐼⟶ℕ0𝑌 Fn 𝐼)
7025, 69syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 Fn 𝐼)
71 eqidd 2611 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) = (𝑌𝑧))
7258, 70, 8, 8, 61, 62, 71offval 6802 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑌) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑌𝑧))))
738, 42, 67, 68, 72ofrfval2 6813 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝑟 ≤ (𝐹𝑓𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
7453, 65, 733bitr4d 299 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
756, 74mpbid 221 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟 ≤ (𝐹𝑓𝑌))
76 breq1 4586 . . . 4 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝐹𝑓𝑌) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7776elrab 3331 . . 3 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7816, 75, 77sylanbrc 695 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)})
7941, 78jca 553 1 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ∘𝑟 cofr 6794   ↑𝑚 cmap 7744  Fincfn 7841  ℝcr 9814   + caddc 9818   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170 This theorem is referenced by:  gsumbagdiag  19197  psrass1lem  19198
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