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Theorem gsummpt2d 32776
Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19927. (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 2728 . . 3 (0g𝑊) = (0g𝑊)
3 gsummpt2d.m . . 3 (𝜑𝑊 ∈ CMnd)
4 gsummpt2d.2 . . 3 (𝜑𝐴 ∈ Fin)
54dmexd 7911 . . 3 (𝜑 → dom 𝐴 ∈ V)
6 gsummpt2d.3 . . 3 ((𝜑𝑥𝐴) → 𝐶𝐵)
7 gsummpt2d.r . . . 4 (𝜑 → Rel 𝐴)
8 1stdm 8044 . . . 4 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
97, 8sylan 579 . . 3 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
10 fo1st 8013 . . . . . 6 1st :V–onto→V
11 fofn 6813 . . . . . 6 (1st :V–onto→V → 1st Fn V)
12 dffn5 6957 . . . . . . 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 5981 . . . 4 (1st𝐴) = ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴)
16 ssv 4004 . . . . 5 𝐴 ⊆ V
17 resmpt 6041 . . . . 5 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥)))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (1st𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (1st𝑥))
1915, 18eqtri 2756 . . 3 (1st𝐴) = (𝑥𝐴 ↦ (1st𝑥))
201, 2, 3, 4, 5, 6, 9, 19gsummpt2co 32775 . 2 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))))
21 gsummpt2d.0 . . . 4 𝑦𝜑
223adantr 480 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
234adantr 480 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24 imaexg 7921 . . . . . . 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 481 . . . . . . . . 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 583 . . . . . . . . 9 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶𝐵)
3127, 30eqeltrrd 2830 . . . . . . . 8 (((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷𝐵)
32 vex 3475 . . . . . . . . . . . 12 𝑦 ∈ V
33 vex 3475 . . . . . . . . . . . 12 𝑧 ∈ V
3432, 33elimasn 6093 . . . . . . . . . . 11 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3534biimpi 215 . . . . . . . . . 10 (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
3635adantl 481 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37 simpr 484 . . . . . . . . . 10 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
3837eqeq1d 2730 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39 eqidd 2729 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
4036, 38, 39rspcedvd 3611 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
4131, 40r19.29a 3159 . . . . . . 7 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷𝐵)
4241fmpttd 7125 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43 eqid 2728 . . . . . . 7 (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44 imafi2 32506 . . . . . . . . 9 (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
454, 44syl 17 . . . . . . . 8 (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
4645adantr 480 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47 fvex 6910 . . . . . . . 8 (0g𝑊) ∈ V
4847a1i 11 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (0g𝑊) ∈ V)
4943, 46, 41, 48fsuppmptdm 9400 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g𝑊))
50 2ndconst 8106 . . . . . . . 8 (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
5150adantl 481 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52 1stpreimas 32498 . . . . . . . . . 10 ((Rel 𝐴𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
537, 52sylan 579 . . . . . . . . 9 ((𝜑𝑦 ∈ dom 𝐴) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
5453reseq2d 5985 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
5554f1oeq1d 6834 . . . . . . 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 19871 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))))
58 simpr 484 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ((1st𝐴) “ {𝑦}))
5953adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → ((1st𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
6058, 59eleqtrd 2831 . . . . . . . . . 10 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
61 xp2nd 8026 . . . . . . . . . 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 8014 . . . . . . . . . . . 12 2nd :V–onto→V
65 fofn 6813 . . . . . . . . . . . 12 (2nd :V–onto→V → 2nd Fn V)
66 dffn5 6957 . . . . . . . . . . . . 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 5981 . . . . . . . . . 10 (2nd ↾ ((1st𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ((1st𝐴) “ {𝑦}))
70 ssv 4004 . . . . . . . . . . 11 ((1st𝐴) “ {𝑦}) ⊆ V
71 resmpt 6041 . . . . . . . . . . 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 2756 . . . . . . . . 9 (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥))
7473a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (2nd ↾ ((1st𝐴) “ {𝑦})) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥)))
75 eqidd 2729 . . . . . . . 8 ((𝜑𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
7663, 74, 75fmptcos 7140 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷))
77 nfv 1910 . . . . . . . . 9 𝑧((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦}))
78 gsummpt2d.c . . . . . . . . . 10 𝑧𝐶
7978a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → 𝑧𝐶)
8060adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
81 xp1st 8025 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st𝑥) ∈ {𝑦})
8280, 81syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) ∈ {𝑦})
83 fvex 6910 . . . . . . . . . . . . . 14 (1st𝑥) ∈ V
8483elsn 4644 . . . . . . . . . . . . 13 ((1st𝑥) ∈ {𝑦} ↔ (1st𝑥) = 𝑦)
8582, 84sylib 217 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (1st𝑥) = 𝑦)
86 simpr 484 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝑧 = (2nd𝑥))
8786eqcomd 2734 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → (2nd𝑥) = 𝑧)
88 eqopi 8029 . . . . . . . . . . . 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 2734 . . . . . . . . 9 ((((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) ∧ 𝑧 = (2nd𝑥)) → 𝐷 = 𝐶)
9277, 79, 62, 91csbiedf 3923 . . . . . . . 8 (((𝜑𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ ((1st𝐴) “ {𝑦})) → (2nd𝑥) / 𝑧𝐷 = 𝐶)
9392mpteq2dva 5248 . . . . . . 7 ((𝜑𝑦 ∈ dom 𝐴) → (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ (2nd𝑥) / 𝑧𝐷) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9476, 93eqtrd 2768 . . . . . 6 ((𝜑𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦}))) = (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))
9594oveq2d 7436 . . . . 5 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ ((1st𝐴) “ {𝑦})))) = (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)))
9657, 95eqtr2d 2769 . . . 4 ((𝜑𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
9721, 96mpteq2da 5246 . . 3 (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ ((1st𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
9897oveq2d 7436 . 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 2768 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 395   = wceq 1534  wnf 1778  wcel 2099  wnfc 2879  Vcvv 3471  csb 3892  wss 3947  {csn 4629  cop 4635  cmpt 5231   × cxp 5676  ccnv 5677  dom cdm 5678  cres 5680  cima 5681  ccom 5682  Rel wrel 5683   Fn wfn 6543  ontowfo 6546  1-1-ontowf1o 6547  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  Fincfn 8964  Basecbs 17180  0gc0g 17421   Σg cgsu 17422  CMndccmn 19735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7871  df-1st 7993  df-2nd 7994  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9387  df-oi 9534  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-fzo 13661  df-seq 14000  df-hash 14323  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-0g 17423  df-gsum 17424  df-mre 17566  df-mrc 17567  df-acs 17569  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18741  df-mulg 19024  df-cntz 19268  df-cmn 19737
This theorem is referenced by:  gsumhashmul  32783  esum2d  33712
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