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Theorem gsummpt2d 31891
Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19749. (Contributed by Thierry Arnoux, 27-Apr-2020.)
Hypotheses
Ref Expression
gsummpt2d.c 𝑧𝐶
gsummpt2d.0 𝑦𝜑
gsummpt2d.b 𝐵 = (Base‘𝑊)
gsummpt2d.1 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
gsummpt2d.r (𝜑 → Rel 𝐴)
gsummpt2d.2 (𝜑𝐴 ∈ Fin)
gsummpt2d.m (𝜑𝑊 ∈ CMnd)
gsummpt2d.3 ((𝜑𝑥𝐴) → 𝐶𝐵)
Assertion
Ref Expression
gsummpt2d (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑦,𝐶   𝑥,𝐷   𝑥,𝑊,𝑦   𝜑,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑥,𝑧)   𝐷(𝑦,𝑧)   𝑊(𝑧)

Proof of Theorem gsummpt2d
StepHypRef Expression
1 gsummpt2d.b . . 3 𝐵 = (Base‘𝑊)
2 eqid 2736 . . 3 (0g𝑊) = (0g𝑊)
3 gsummpt2d.m . . 3 (𝜑𝑊 ∈ CMnd)
4 gsummpt2d.2 . . 3 (𝜑𝐴 ∈ Fin)
54dmexd 7842 . . 3 (𝜑 → dom 𝐴 ∈ V)
6 gsummpt2d.3 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
7 gsummpt2d.r . . . 4 (𝜑 → Rel 𝐴)
8 1stdm 7972 . . . 4 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
97, 8sylan 580 . . 3 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
10 fo1st 7941 . . . . . 6 1st :V–onto→V
11 fofn 6758 . . . . . 6 (1st :V–onto→V → 1st Fn V)
12 dffn5 6901 . . . . . . 7 (1st Fn V ↔ 1st = (𝑥 ∈ V ↦ (1st𝑥)))
1312biimpi 215 . . . . . 6 (1st Fn V → 1st = (𝑥 ∈ V ↦ (1st𝑥)))
1410, 11, 13mp2b 10 . . . . 5 1st = (𝑥 ∈ V ↦ (1st𝑥))
1514reseq1i 5933 . . . 4 (1st𝐴) = ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴)
16 ssv 3968 . . . . 5 𝐴 ⊆ V
17 resmpt 5991 . . . . 5 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥)))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥))
1915, 18eqtri 2764 . . 3 (1st𝐴) = (𝑥𝐴 ↦ (1st𝑥))
201, 2, 3, 4, 5, 6, 9, 19gsummpt2co 31890 . 2 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))))
21 gsummpt2d.0 . . . 4 𝑦𝜑
223adantr 481 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
234adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24 imaexg 7852 . . . . . . 7 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ V)
2523, 24syl 17 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ V)
26 gsummpt2d.1 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
2726adantl 482 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 = 𝐷)
28 simp-4l 781 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝜑)
29 simplr 767 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥𝐴)
3028, 29, 6syl2anc 584 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶𝐵)
3127, 30eqeltrrd 2839 . . . . . . . 8 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷𝐵)
32 vex 3449 . . . . . . . . . . . 12 𝑦 ∈ V
33 vex 3449 . . . . . . . . . . . 12 𝑧 ∈ V
3432, 33elimasn 6041 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3534biimpi 215 . . . . . . . . . 10 (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3635adantl 482 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37 simpr 485 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
3837eqeq1d 2738 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39 eqidd 2737 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
4036, 38, 39rspcedvd 3583 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
4131, 40r19.29a 3159 . . . . . . 7 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷𝐵)
4241fmpttd 7063 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43 eqid 2736 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44 imafi2 31628 . . . . . . . . 9 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
454, 44syl 17 . . . . . . . 8 (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
4645adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47 fvex 6855 . . . . . . . 8 (0g𝑊) ∈ V
4847a1i 11 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (0g𝑊) ∈ V)
4943, 46, 41, 48fsuppmptdm 9316 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g𝑊))
50 2ndconst 8033 . . . . . . . 8 (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
5150adantl 482 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52 1stpreimas 31619 . . . . . . . . . 10 ((Rel 𝐴𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
537, 52sylan 580 . . . . . . . . 9 ((𝜑𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
5453reseq2d 5937 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
5554f1oeq1d 6779 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → ((2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}) ↔ (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦})))
5651, 55mpbird 256 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
571, 2, 22, 25, 42, 49, 56gsumf1o 19693 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))))
58 simpr 485 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ((1st𝐴) “ {𝑦}))
5953adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
6058, 59eleqtrd 2840 . . . . . . . . . 10 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
61 xp2nd 7954 . . . . . . . . . 10 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6260, 61syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6362ralrimiva 3143 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ ((1st𝐴) “ {𝑦})(2nd𝑥) ∈ (𝐴 “ {𝑦}))
64 fo2nd 7942 . . . . . . . . . . . 12 2nd :V–onto→V
65 fofn 6758 . . . . . . . . . . . 12 (2nd :V–onto→V → 2nd Fn V)
66 dffn5 6901 . . . . . . . . . . . . 13 (2nd Fn V ↔ 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
6766biimpi 215 . . . . . . . . . . . 12 (2nd Fn V → 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
6864, 65, 67mp2b 10 . . . . . . . . . . 11 2nd = (𝑥 ∈ V ↦ (2nd𝑥))
6968reseq1i 5933 . . . . . . . . . 10 (2nd ↾ ((1st𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦}))
70 ssv 3968 . . . . . . . . . . 11 ((1st𝐴) “ {𝑦}) ⊆ V
71 resmpt 5991 . . . . . . . . . . 11 (((1st𝐴) “ {𝑦}) ⊆ V → ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
7270, 71ax-mp 5 . . . . . . . . . 10 ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7369, 72eqtri 2764 . . . . . . . . 9 (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7473a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
75 eqidd 2737 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
7663, 74, 75fmptcos 7077 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷))
77 nfv 1917 . . . . . . . . 9 𝑧((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦}))
78 gsummpt2d.c . . . . . . . . . 10 𝑧𝐶
7978a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑧𝐶)
8060adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
81 xp1st 7953 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st𝑥) ∈ {𝑦})
8280, 81syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) ∈ {𝑦})
83 fvex 6855 . . . . . . . . . . . . . 14 (1st𝑥) ∈ V
8483elsn 4601 . . . . . . . . . . . . 13 ((1st𝑥) ∈ {𝑦} ↔ (1st𝑥) = 𝑦)
8582, 84sylib 217 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) = 𝑦)
86 simpr 485 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑧 = (2nd𝑥))
8786eqcomd 2742 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (2nd𝑥) = 𝑧)
88 eqopi 7957 . . . . . . . . . . . 12 ((𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝑧)) → 𝑥 = ⟨𝑦, 𝑧⟩)
8980, 85, 87, 88syl12anc 835 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 = ⟨𝑦, 𝑧⟩)
9089, 26syl 17 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐶 = 𝐷)
9190eqcomd 2742 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐷 = 𝐶)
9277, 79, 62, 91csbiedf 3886 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) / 𝑧𝐷 = 𝐶)
9392mpteq2dva 5205 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9476, 93eqtrd 2776 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9594oveq2d 7373 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))) = (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))
9657, 95eqtr2d 2777 . . . 4 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
9721, 96mpteq2da 5203 . . 3 (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
9897oveq2d 7373 . 2 (𝜑 → (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
9920, 98eqtrd 2776 1 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  wnfc 2887  Vcvv 3445  csb 3855  wss 3910  {csn 4586  cop 4592  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  cres 5635  cima 5636  ccom 5637  Rel wrel 5638   Fn wfn 6491  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Fincfn 8883  Basecbs 17083  0gc0g 17321   Σg cgsu 17322  CMndccmn 19562
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564
This theorem is referenced by:  gsumhashmul  31898  esum2d  32692
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