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Theorem gsummpt2d 31940
Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19754. (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 2733 . . 3 (0g𝑊) = (0g𝑊)
3 gsummpt2d.m . . 3 (𝜑𝑊 ∈ CMnd)
4 gsummpt2d.2 . . 3 (𝜑𝐴 ∈ Fin)
54dmexd 7843 . . 3 (𝜑 → dom 𝐴 ∈ V)
6 gsummpt2d.3 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
7 gsummpt2d.r . . . 4 (𝜑 → Rel 𝐴)
8 1stdm 7973 . . . 4 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
97, 8sylan 581 . . 3 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
10 fo1st 7942 . . . . . 6 1st :V–onto→V
11 fofn 6759 . . . . . 6 (1st :V–onto→V → 1st Fn V)
12 dffn5 6902 . . . . . . 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 5934 . . . 4 (1st𝐴) = ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴)
16 ssv 3969 . . . . 5 𝐴 ⊆ V
17 resmpt 5992 . . . . 5 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥)))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥))
1915, 18eqtri 2761 . . 3 (1st𝐴) = (𝑥𝐴 ↦ (1st𝑥))
201, 2, 3, 4, 5, 6, 9, 19gsummpt2co 31939 . 2 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))))
21 gsummpt2d.0 . . . 4 𝑦𝜑
223adantr 482 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
234adantr 482 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24 imaexg 7853 . . . . . . 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 483 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 = 𝐷)
28 simp-4l 782 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝜑)
29 simplr 768 . . . . . . . . . 10 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥𝐴)
3028, 29, 6syl2anc 585 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶𝐵)
3127, 30eqeltrrd 2835 . . . . . . . 8 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷𝐵)
32 vex 3448 . . . . . . . . . . . 12 𝑦 ∈ V
33 vex 3448 . . . . . . . . . . . 12 𝑧 ∈ V
3432, 33elimasn 6042 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3534biimpi 215 . . . . . . . . . 10 (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3635adantl 483 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37 simpr 486 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
3837eqeq1d 2735 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39 eqidd 2734 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
4036, 38, 39rspcedvd 3582 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
4131, 40r19.29a 3156 . . . . . . 7 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷𝐵)
4241fmpttd 7064 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43 eqid 2733 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44 imafi2 31675 . . . . . . . . 9 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
454, 44syl 17 . . . . . . . 8 (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
4645adantr 482 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47 fvex 6856 . . . . . . . 8 (0g𝑊) ∈ V
4847a1i 11 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (0g𝑊) ∈ V)
4943, 46, 41, 48fsuppmptdm 9321 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g𝑊))
50 2ndconst 8034 . . . . . . . 8 (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
5150adantl 483 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52 1stpreimas 31666 . . . . . . . . . 10 ((Rel 𝐴𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
537, 52sylan 581 . . . . . . . . 9 ((𝜑𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
5453reseq2d 5938 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
5554f1oeq1d 6780 . . . . . . 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 257 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
571, 2, 22, 25, 42, 49, 56gsumf1o 19698 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))))
58 simpr 486 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ((1st𝐴) “ {𝑦}))
5953adantr 482 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
6058, 59eleqtrd 2836 . . . . . . . . . 10 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
61 xp2nd 7955 . . . . . . . . . 10 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6260, 61syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) ∈ (𝐴 “ {𝑦}))
6362ralrimiva 3140 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ ((1st𝐴) “ {𝑦})(2nd𝑥) ∈ (𝐴 “ {𝑦}))
64 fo2nd 7943 . . . . . . . . . . . 12 2nd :V–onto→V
65 fofn 6759 . . . . . . . . . . . 12 (2nd :V–onto→V → 2nd Fn V)
66 dffn5 6902 . . . . . . . . . . . . 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 5934 . . . . . . . . . 10 (2nd ↾ ((1st𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦}))
70 ssv 3969 . . . . . . . . . . 11 ((1st𝐴) “ {𝑦}) ⊆ V
71 resmpt 5992 . . . . . . . . . . 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 2761 . . . . . . . . 9 (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7473a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
75 eqidd 2734 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
7663, 74, 75fmptcos 7078 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷))
77 nfv 1918 . . . . . . . . 9 𝑧((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦}))
78 gsummpt2d.c . . . . . . . . . 10 𝑧𝐶
7978a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑧𝐶)
8060adantr 482 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
81 xp1st 7954 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st𝑥) ∈ {𝑦})
8280, 81syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) ∈ {𝑦})
83 fvex 6856 . . . . . . . . . . . . . 14 (1st𝑥) ∈ V
8483elsn 4602 . . . . . . . . . . . . 13 ((1st𝑥) ∈ {𝑦} ↔ (1st𝑥) = 𝑦)
8582, 84sylib 217 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) = 𝑦)
86 simpr 486 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑧 = (2nd𝑥))
8786eqcomd 2739 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (2nd𝑥) = 𝑧)
88 eqopi 7958 . . . . . . . . . . . 12 ((𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝑧)) → 𝑥 = ⟨𝑦, 𝑧⟩)
8980, 85, 87, 88syl12anc 836 . . . . . . . . . . 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 2739 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐷 = 𝐶)
9277, 79, 62, 91csbiedf 3887 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) / 𝑧𝐷 = 𝐶)
9392mpteq2dva 5206 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9476, 93eqtrd 2773 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9594oveq2d 7374 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))) = (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))
9657, 95eqtr2d 2774 . . . 4 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
9721, 96mpteq2da 5204 . . 3 (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
9897oveq2d 7374 . 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 2773 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 397   = wceq 1542  wnf 1786  wcel 2107  wnfc 2884  Vcvv 3444  csb 3856  wss 3911  {csn 4587  cop 4593  cmpt 5189   × cxp 5632  ccnv 5633  dom cdm 5634  cres 5636  cima 5637  ccom 5638  Rel wrel 5639   Fn wfn 6492  ontowfo 6495  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Fincfn 8886  Basecbs 17088  0gc0g 17326   Σg cgsu 17327  CMndccmn 19567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-seq 13913  df-hash 14237  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-0g 17328  df-gsum 17329  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-mulg 18878  df-cntz 19102  df-cmn 19569
This theorem is referenced by:  gsumhashmul  31947  esum2d  32749
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