Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumzaddlem Structured version   Visualization version   GIF version

 Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
Hypotheses
Ref Expression
gsumzaddlem.w 𝑊 = ((𝐹𝐻) supp 0 )
gsumzaddlem.1 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzaddlem.2 (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
gsumzaddlem.3 (𝜑 → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))
gsumzaddlem.4 ((𝜑 ∧ (𝑥𝐴𝑘 ∈ (𝐴𝑥))) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))
Assertion
Ref Expression
gsumzaddlem (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Distinct variable groups:   𝑥,𝑘, +   0 ,𝑘,𝑥   𝑘,𝐹,𝑥   𝑘,𝐺,𝑥   𝐴,𝑘,𝑥   𝐵,𝑘,𝑥   𝑘,𝐻,𝑥   𝜑,𝑘,𝑥   𝑥,𝑉   𝑘,𝑊,𝑥   𝑘,𝑍,𝑥
Allowed substitution hint:   𝑉(𝑘)

Dummy variables 𝑓 𝑛 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzadd.g . . . . . 6 (𝜑𝐺 ∈ Mnd)
2 gsumzadd.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 gsumzadd.0 . . . . . . . 8 0 = (0g𝐺)
42, 3mndidcl 17926 . . . . . . 7 (𝐺 ∈ Mnd → 0𝐵)
51, 4syl 17 . . . . . 6 (𝜑0𝐵)
6 gsumzadd.p . . . . . . 7 + = (+g𝐺)
72, 6, 3mndlid 17931 . . . . . 6 ((𝐺 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
81, 5, 7syl2anc 587 . . . . 5 (𝜑 → ( 0 + 0 ) = 0 )
98adantr 484 . . . 4 ((𝜑𝑊 = ∅) → ( 0 + 0 ) = 0 )
10 gsumzaddlem.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
11 gsumzadd.a . . . . . . . 8 (𝜑𝐴𝑉)
123fvexi 6675 . . . . . . . . 9 0 ∈ V
1312a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
14 gsumzaddlem.h . . . . . . . . . . 11 (𝜑𝐻:𝐴𝐵)
15 fex 6980 . . . . . . . . . . 11 ((𝐻:𝐴𝐵𝐴𝑉) → 𝐻 ∈ V)
1614, 11, 15syl2anc 587 . . . . . . . . . 10 (𝜑𝐻 ∈ V)
1716suppun 7846 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
18 gsumzaddlem.w . . . . . . . . 9 𝑊 = ((𝐹𝐻) supp 0 )
1917, 18sseqtrrdi 4004 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
2010, 11, 13, 19gsumcllem 19028 . . . . . . 7 ((𝜑𝑊 = ∅) → 𝐹 = (𝑥𝐴0 ))
2120oveq2d 7165 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐴0 )))
223gsumz 18000 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑥𝐴0 )) = 0 )
231, 11, 22syl2anc 587 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑥𝐴0 )) = 0 )
2423adantr 484 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝑥𝐴0 )) = 0 )
2521, 24eqtrd 2859 . . . . 5 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐹) = 0 )
26 fex 6980 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2710, 11, 26syl2anc 587 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
2827suppun 7846 . . . . . . . . . 10 (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐻𝐹) supp 0 ))
29 uncom 4115 . . . . . . . . . . 11 (𝐹𝐻) = (𝐻𝐹)
3029oveq1i 7159 . . . . . . . . . 10 ((𝐹𝐻) supp 0 ) = ((𝐻𝐹) supp 0 )
3128, 30sseqtrrdi 4004 . . . . . . . . 9 (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
3231, 18sseqtrrdi 4004 . . . . . . . 8 (𝜑 → (𝐻 supp 0 ) ⊆ 𝑊)
3314, 11, 13, 32gsumcllem 19028 . . . . . . 7 ((𝜑𝑊 = ∅) → 𝐻 = (𝑥𝐴0 ))
3433oveq2d 7165 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐻) = (𝐺 Σg (𝑥𝐴0 )))
3534, 24eqtrd 2859 . . . . 5 ((𝜑𝑊 = ∅) → (𝐺 Σg 𝐻) = 0 )
3625, 35oveq12d 7167 . . . 4 ((𝜑𝑊 = ∅) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ( 0 + 0 ))
3711adantr 484 . . . . . . . 8 ((𝜑𝑊 = ∅) → 𝐴𝑉)
385ad2antrr 725 . . . . . . . 8 (((𝜑𝑊 = ∅) ∧ 𝑥𝐴) → 0𝐵)
3937, 38, 38, 20, 33offval2 7420 . . . . . . 7 ((𝜑𝑊 = ∅) → (𝐹f + 𝐻) = (𝑥𝐴 ↦ ( 0 + 0 )))
409mpteq2dv 5148 . . . . . . 7 ((𝜑𝑊 = ∅) → (𝑥𝐴 ↦ ( 0 + 0 )) = (𝑥𝐴0 ))
4139, 40eqtrd 2859 . . . . . 6 ((𝜑𝑊 = ∅) → (𝐹f + 𝐻) = (𝑥𝐴0 ))
4241oveq2d 7165 . . . . 5 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝐹f + 𝐻)) = (𝐺 Σg (𝑥𝐴0 )))
4342, 24eqtrd 2859 . . . 4 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝐹f + 𝐻)) = 0 )
449, 36, 433eqtr4rd 2870 . . 3 ((𝜑𝑊 = ∅) → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
4544ex 416 . 2 (𝜑 → (𝑊 = ∅ → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
461adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐺 ∈ Mnd)
472, 6mndcl 17919 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ 𝑧𝐵𝑤𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
48473expb 1117 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 + 𝑤) ∈ 𝐵)
4946, 48sylan 583 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 + 𝑤) ∈ 𝐵)
5049caovclg 7334 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
51 simprl 770 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ ℕ)
52 nnuz 12278 . . . . . . . 8 ℕ = (ℤ‘1)
5351, 52eleqtrdi 2926 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ (ℤ‘1))
5410adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹:𝐴𝐵)
55 f1of1 6605 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–1-1𝑊)
5655ad2antll 728 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1𝑊)
57 suppssdm 7839 . . . . . . . . . . . . . 14 ((𝐹𝐻) supp 0 ) ⊆ dom (𝐹𝐻)
5857a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝐻) supp 0 ) ⊆ dom (𝐹𝐻))
5918a1i 11 . . . . . . . . . . . . 13 (𝜑𝑊 = ((𝐹𝐻) supp 0 ))
60 dmun 5766 . . . . . . . . . . . . . 14 dom (𝐹𝐻) = (dom 𝐹 ∪ dom 𝐻)
6110fdmd 6513 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐹 = 𝐴)
6214fdmd 6513 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐻 = 𝐴)
6361, 62uneq12d 4126 . . . . . . . . . . . . . . 15 (𝜑 → (dom 𝐹 ∪ dom 𝐻) = (𝐴𝐴))
64 unidm 4114 . . . . . . . . . . . . . . 15 (𝐴𝐴) = 𝐴
6563, 64syl6eq 2875 . . . . . . . . . . . . . 14 (𝜑 → (dom 𝐹 ∪ dom 𝐻) = 𝐴)
6660, 65syl5req 2872 . . . . . . . . . . . . 13 (𝜑𝐴 = dom (𝐹𝐻))
6758, 59, 663sstr4d 4000 . . . . . . . . . . . 12 (𝜑𝑊𝐴)
6867adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
69 f1ss 6571 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝑊))–1-1𝑊𝑊𝐴) → 𝑓:(1...(♯‘𝑊))–1-1𝐴)
7056, 68, 69syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1𝐴)
71 f1f 6565 . . . . . . . . . 10 (𝑓:(1...(♯‘𝑊))–1-1𝐴𝑓:(1...(♯‘𝑊))⟶𝐴)
7270, 71syl 17 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝐴)
73 fco 6521 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹𝑓):(1...(♯‘𝑊))⟶𝐵)
7454, 72, 73syl2anc 587 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹𝑓):(1...(♯‘𝑊))⟶𝐵)
7574ffvelrnda 6842 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
7614adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐻:𝐴𝐵)
77 fco 6521 . . . . . . . . 9 ((𝐻:𝐴𝐵𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻𝑓):(1...(♯‘𝑊))⟶𝐵)
7876, 72, 77syl2anc 587 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻𝑓):(1...(♯‘𝑊))⟶𝐵)
7978ffvelrnda 6842 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘𝑘) ∈ 𝐵)
8054ffnd 6504 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
8176ffnd 6504 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐻 Fn 𝐴)
8211adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐴𝑉)
83 ovexd 7184 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (1...(♯‘𝑊)) ∈ V)
84 inidm 4180 . . . . . . . . . . 11 (𝐴𝐴) = 𝐴
8580, 81, 72, 82, 82, 83, 84ofco 7423 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐹f + 𝐻) ∘ 𝑓) = ((𝐹𝑓) ∘f + (𝐻𝑓)))
8685fveq1d 6663 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (((𝐹f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹𝑓) ∘f + (𝐻𝑓))‘𝑘))
8786adantr 484 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹𝑓) ∘f + (𝐻𝑓))‘𝑘))
88 fnfco 6533 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹𝑓) Fn (1...(♯‘𝑊)))
8980, 72, 88syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹𝑓) Fn (1...(♯‘𝑊)))
90 fnfco 6533 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻𝑓) Fn (1...(♯‘𝑊)))
9181, 72, 90syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻𝑓) Fn (1...(♯‘𝑊)))
92 inidm 4180 . . . . . . . . 9 ((1...(♯‘𝑊)) ∩ (1...(♯‘𝑊))) = (1...(♯‘𝑊))
93 eqidd 2825 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = ((𝐹𝑓)‘𝑘))
94 eqidd 2825 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘𝑘) = ((𝐻𝑓)‘𝑘))
9589, 91, 83, 83, 92, 93, 94ofval 7412 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹𝑓) ∘f + (𝐻𝑓))‘𝑘) = (((𝐹𝑓)‘𝑘) + ((𝐻𝑓)‘𝑘)))
9687, 95eqtrd 2859 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹𝑓)‘𝑘) + ((𝐻𝑓)‘𝑘)))
971ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐺 ∈ Mnd)
98 elfzouz 13046 . . . . . . . . . 10 (𝑛 ∈ (1..^(♯‘𝑊)) → 𝑛 ∈ (ℤ‘1))
9998adantl 485 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ (ℤ‘1))
100 elfzouz2 13056 . . . . . . . . . . . . 13 (𝑛 ∈ (1..^(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ𝑛))
101100adantl 485 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (♯‘𝑊) ∈ (ℤ𝑛))
102 fzss2 12951 . . . . . . . . . . . 12 ((♯‘𝑊) ∈ (ℤ𝑛) → (1...𝑛) ⊆ (1...(♯‘𝑊)))
103101, 102syl 17 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (1...𝑛) ⊆ (1...(♯‘𝑊)))
104103sselda 3953 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...(♯‘𝑊)))
10575adantlr 714 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
106104, 105syldan 594 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐹𝑓)‘𝑘) ∈ 𝐵)
1072, 6mndcl 17919 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑘𝐵𝑥𝐵) → (𝑘 + 𝑥) ∈ 𝐵)
1081073expb 1117 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑘𝐵𝑥𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
10997, 108sylan 583 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ (𝑘𝐵𝑥𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
11099, 106, 109seqcl 13395 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐹𝑓))‘𝑛) ∈ 𝐵)
11179adantlr 714 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘𝑘) ∈ 𝐵)
112104, 111syldan 594 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐻𝑓)‘𝑘) ∈ 𝐵)
11399, 112, 109seqcl 13395 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻𝑓))‘𝑛) ∈ 𝐵)
114 fzofzp1 13138 . . . . . . . . 9 (𝑛 ∈ (1..^(♯‘𝑊)) → (𝑛 + 1) ∈ (1...(♯‘𝑊)))
115 ffvelrn 6840 . . . . . . . . 9 (((𝐹𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) ∈ 𝐵)
11674, 114, 115syl2an 598 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) ∈ 𝐵)
117 ffvelrn 6840 . . . . . . . . 9 (((𝐻𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐻𝑓)‘(𝑛 + 1)) ∈ 𝐵)
11878, 114, 117syl2an 598 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻𝑓)‘(𝑛 + 1)) ∈ 𝐵)
119 fvco3 6751 . . . . . . . . . . . 12 ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1))))
12072, 114, 119syl2an 598 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1))))
121 fveq2 6661 . . . . . . . . . . . . 13 (𝑘 = (𝑓‘(𝑛 + 1)) → (𝐹𝑘) = (𝐹‘(𝑓‘(𝑛 + 1))))
122121eleq1d 2900 . . . . . . . . . . . 12 (𝑘 = (𝑓‘(𝑛 + 1)) → ((𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ↔ (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
123 gsumzaddlem.4 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑥𝐴𝑘 ∈ (𝐴𝑥))) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))
124123expr 460 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑘 ∈ (𝐴𝑥) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
125124ralrimiv 3176 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))
126125ex 416 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
127126alrimiv 1929 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
128127ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑥(𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})))
129 imassrn 5927 . . . . . . . . . . . . . 14 (𝑓 “ (1...𝑛)) ⊆ ran 𝑓
13072adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))⟶𝐴)
131130frnd 6510 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝑓𝐴)
132129, 131sstrid 3964 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ⊆ 𝐴)
133 vex 3483 . . . . . . . . . . . . . . 15 𝑓 ∈ V
134133imaex 7616 . . . . . . . . . . . . . 14 (𝑓 “ (1...𝑛)) ∈ V
135 sseq1 3978 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓 “ (1...𝑛)) → (𝑥𝐴 ↔ (𝑓 “ (1...𝑛)) ⊆ 𝐴))
136 difeq2 4079 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓 “ (1...𝑛)) → (𝐴𝑥) = (𝐴 ∖ (𝑓 “ (1...𝑛))))
137 reseq2 5835 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑓 “ (1...𝑛)) → (𝐻𝑥) = (𝐻 ↾ (𝑓 “ (1...𝑛))))
138137oveq2d 7165 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑓 “ (1...𝑛)) → (𝐺 Σg (𝐻𝑥)) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
139138sneqd 4562 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑓 “ (1...𝑛)) → {(𝐺 Σg (𝐻𝑥))} = {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})
140139fveq2d 6665 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑓 “ (1...𝑛)) → (𝑍‘{(𝐺 Σg (𝐻𝑥))}) = (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
141140eleq2d 2901 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓 “ (1...𝑛)) → ((𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}) ↔ (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
142136, 141raleqbidv 3392 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓 “ (1...𝑛)) → (∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}) ↔ ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
143135, 142imbi12d 348 . . . . . . . . . . . . . 14 (𝑥 = (𝑓 “ (1...𝑛)) → ((𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})) ↔ ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))))
144134, 143spcv 3592 . . . . . . . . . . . . 13 (∀𝑥(𝑥𝐴 → ∀𝑘 ∈ (𝐴𝑥)(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))})) → ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))
145128, 132, 144sylc 65 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
146 ffvelrn 6840 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴)
14772, 114, 146syl2an 598 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴)
148 fzp1nel 12995 . . . . . . . . . . . . . 14 ¬ (𝑛 + 1) ∈ (1...𝑛)
14970adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))–1-1𝐴)
150114adantl 485 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑛 + 1) ∈ (1...(♯‘𝑊)))
151 f1elima 7013 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝑊))–1-1𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊)) ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛)))
152149, 150, 103, 151syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛)))
153148, 152mtbiri 330 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ¬ (𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)))
154147, 153eldifd 3930 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ (𝐴 ∖ (𝑓 “ (1...𝑛))))
155122, 145, 154rspcdva 3611 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
156120, 155eqeltrd 2916 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))
157 gsumzadd.z . . . . . . . . . . . . 13 𝑍 = (Cntz‘𝐺)
158134a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ∈ V)
15914ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐻:𝐴𝐵)
160159, 132fssresd 6535 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐻 ↾ (𝑓 “ (1...𝑛))):(𝑓 “ (1...𝑛))⟶𝐵)
161 gsumzaddlem.2 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
162161ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
163 resss 5865 . . . . . . . . . . . . . . 15 (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ 𝐻
164163rnssi 5797 . . . . . . . . . . . . . 14 ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻
165157cntzidss 18468 . . . . . . . . . . . . . 14 ((ran 𝐻 ⊆ (𝑍‘ran 𝐻) ∧ ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛)))))
166162, 164, 165sylancl 589 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛)))))
16799, 52eleqtrrdi 2927 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ ℕ)
168 f1ores 6620 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝑊))–1-1𝐴 ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)))
169149, 103, 168syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)))
170 f1of1 6605 . . . . . . . . . . . . . 14 ((𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛)))
171169, 170syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛)))
172 suppssdm 7839 . . . . . . . . . . . . . . 15 ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ dom (𝐻 ↾ (𝑓 “ (1...𝑛)))
173 dmres 5862 . . . . . . . . . . . . . . . 16 dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻)
174173a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻))
175172, 174sseqtrid 4005 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ((𝑓 “ (1...𝑛)) ∩ dom 𝐻))
176 inss1 4190 . . . . . . . . . . . . . . 15 ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ (𝑓 “ (1...𝑛))
177 df-ima 5555 . . . . . . . . . . . . . . . 16 (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛))
178177a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛)))
179176, 178sseqtrid 4005 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ ran (𝑓 ↾ (1...𝑛)))
180175, 179sstrd 3963 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ran (𝑓 ↾ (1...𝑛)))
181 eqid 2824 . . . . . . . . . . . . 13 (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 ) = (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 )
1822, 3, 6, 157, 97, 158, 160, 166, 167, 171, 180, 181gsumval3 19027 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))) = (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛))
183177eqimss2i 4012 . . . . . . . . . . . . . . . . . 18 ran (𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛))
184 cores 6089 . . . . . . . . . . . . . . . . . 18 (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛)) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛))))
185183, 184ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))
186 resco 6090 . . . . . . . . . . . . . . . . 17 ((𝐻𝑓) ↾ (1...𝑛)) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))
187185, 186eqtr4i 2850 . . . . . . . . . . . . . . . 16 ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = ((𝐻𝑓) ↾ (1...𝑛))
188187fveq1i 6662 . . . . . . . . . . . . . . 15 (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = (((𝐻𝑓) ↾ (1...𝑛))‘𝑘)
189 fvres 6680 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → (((𝐻𝑓) ↾ (1...𝑛))‘𝑘) = ((𝐻𝑓)‘𝑘))
190188, 189syl5eq 2871 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑛) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻𝑓)‘𝑘))
191190adantl 485 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻𝑓)‘𝑘))
19299, 191seqfveq 13399 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛) = (seq1( + , (𝐻𝑓))‘𝑛))
193182, 192eqtr2d 2860 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
194 fvex 6674 . . . . . . . . . . . 12 (seq1( + , (𝐻𝑓))‘𝑛) ∈ V
195194elsn 4565 . . . . . . . . . . 11 ((seq1( + , (𝐻𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))} ↔ (seq1( + , (𝐻𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))))
196193, 195sylibr 237 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})
1976, 157cntzi 18459 . . . . . . . . . 10 ((((𝐹𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ∧ (seq1( + , (𝐻𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) → (((𝐹𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻𝑓))‘𝑛)) = ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))))
198156, 196, 197syl2anc 587 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((𝐹𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻𝑓))‘𝑛)) = ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))))
199198eqcomd 2830 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))) = (((𝐹𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻𝑓))‘𝑛)))
2002, 6, 97, 110, 113, 116, 118, 199mnd4g 17925 . . . . . . 7 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((seq1( + , (𝐹𝑓))‘𝑛) + (seq1( + , (𝐻𝑓))‘𝑛)) + (((𝐹𝑓)‘(𝑛 + 1)) + ((𝐻𝑓)‘(𝑛 + 1)))) = (((seq1( + , (𝐹𝑓))‘𝑛) + ((𝐹𝑓)‘(𝑛 + 1))) + ((seq1( + , (𝐻𝑓))‘𝑛) + ((𝐻𝑓)‘(𝑛 + 1)))))
20150, 50, 53, 75, 79, 96, 200seqcaopr3 13410 . . . . . 6 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (seq1( + , ((𝐹f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)) = ((seq1( + , (𝐹𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻𝑓))‘(♯‘𝑊))))
20249, 54, 76, 82, 82, 84off 7418 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹f + 𝐻):𝐴𝐵)
203 gsumzaddlem.3 . . . . . . . 8 (𝜑 → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))
204203adantr 484 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))
20546, 108sylan 583 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑘𝐵𝑥𝐵)) → (𝑘 + 𝑥) ∈ 𝐵)
206205, 54, 76, 82, 82, 84off 7418 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹f + 𝐻):𝐴𝐵)
207 eldifi 4089 . . . . . . . . . 10 (𝑥 ∈ (𝐴 ∖ ran 𝑓) → 𝑥𝐴)
208 eqidd 2825 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
209 eqidd 2825 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥𝐴) → (𝐻𝑥) = (𝐻𝑥))
21080, 81, 82, 82, 84, 208, 209ofval 7412 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥𝐴) → ((𝐹f + 𝐻)‘𝑥) = ((𝐹𝑥) + (𝐻𝑥)))
211207, 210sylan2 595 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹f + 𝐻)‘𝑥) = ((𝐹𝑥) + (𝐻𝑥)))
21217adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
213 f1ofo 6613 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–onto𝑊)
214 forn 6584 . . . . . . . . . . . . . . . 16 (𝑓:(1...(♯‘𝑊))–onto𝑊 → ran 𝑓 = 𝑊)
215213, 214syl 17 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ran 𝑓 = 𝑊)
216215, 18syl6eq 2875 . . . . . . . . . . . . . 14 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ran 𝑓 = ((𝐹𝐻) supp 0 ))
217216sseq2d 3985 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
218217ad2antll 728 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
219212, 218mpbird 260 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐹 supp 0 ) ⊆ ran 𝑓)
22012a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → 0 ∈ V)
22154, 219, 82, 220suppssr 7857 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐹𝑥) = 0 )
22228adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐻𝐹) supp 0 ))
223222, 30sseqtrrdi 4004 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 ))
224216sseq2d 3985 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
225224ad2antll 728 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹𝐻) supp 0 )))
226223, 225mpbird 260 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐻 supp 0 ) ⊆ ran 𝑓)
22776, 226, 82, 220suppssr 7857 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐻𝑥) = 0 )
228221, 227oveq12d 7167 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹𝑥) + (𝐻𝑥)) = ( 0 + 0 ))
2298ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ( 0 + 0 ) = 0 )
230211, 228, 2293eqtrd 2863 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹f + 𝐻)‘𝑥) = 0 )
231206, 230suppss 7856 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐹f + 𝐻) supp 0 ) ⊆ ran 𝑓)
232 ovex 7182 . . . . . . . . 9 (𝐹f + 𝐻) ∈ V
233232, 133coex 7630 . . . . . . . 8 ((𝐹f + 𝐻) ∘ 𝑓) ∈ V
234 suppimacnv 7837 . . . . . . . . 9 ((((𝐹f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (((𝐹f + 𝐻) ∘ 𝑓) supp 0 ) = (((𝐹f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })))
235234eqcomd 2830 . . . . . . . 8 ((((𝐹f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (((𝐹f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹f + 𝐻) ∘ 𝑓) supp 0 ))
236233, 12, 235mp2an 691 . . . . . . 7 (((𝐹f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹f + 𝐻) ∘ 𝑓) supp 0 )
2372, 3, 6, 157, 46, 82, 202, 204, 51, 70, 231, 236gsumval3 19027 . . . . . 6 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg (𝐹f + 𝐻)) = (seq1( + , ((𝐹f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)))
238 gsumzaddlem.1 . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
239238adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
240 eqid 2824 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
2412, 3, 6, 157, 46, 82, 54, 239, 51, 70, 219, 240gsumval3 19027 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))
242161adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐻 ⊆ (𝑍‘ran 𝐻))
243 eqid 2824 . . . . . . . 8 ((𝐻𝑓) supp 0 ) = ((𝐻𝑓) supp 0 )
2442, 3, 6, 157, 46, 82, 76, 242, 51, 70, 226, 243gsumval3 19027 . . . . . . 7 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg 𝐻) = (seq1( + , (𝐻𝑓))‘(♯‘𝑊)))
245241, 244oveq12d 7167 . . . . . 6 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ((seq1( + , (𝐹𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻𝑓))‘(♯‘𝑊))))
246201, 237, 2453eqtr4d 2869 . . . . 5 ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
247246expr 460 . . . 4 ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
248247exlimdv 1935 . . 3 ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
249248expimpd 457 . 2 (𝜑 → (((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
250 gsumzadd.fn . . . . 5 (𝜑𝐹 finSupp 0 )
251 gsumzadd.hn . . . . 5 (𝜑𝐻 finSupp 0 )
252250, 251fsuppun 8849 . . . 4 (𝜑 → ((𝐹𝐻) supp 0 ) ∈ Fin)
25318, 252eqeltrid 2920 . . 3 (𝜑𝑊 ∈ Fin)
254 fz1f1o 15067 . . 3 (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
255253, 254syl 17 . 2 (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)))
25645, 249, 255mpjaod 857 1 (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ∖ cdif 3916   ∪ cun 3917   ∩ cin 3918   ⊆ wss 3919  ∅c0 4276  {csn 4550   class class class wbr 5052   ↦ cmpt 5132  ◡ccnv 5541  dom cdm 5542  ran crn 5543   ↾ cres 5544   “ cima 5545   ∘ ccom 5546   Fn wfn 6338  ⟶wf 6339  –1-1→wf1 6340  –onto→wfo 6341  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7149   ∘f cof 7401   supp csupp 7826  Fincfn 8505   finSupp cfsupp 8830  1c1 10536   + caddc 10538  ℕcn 11634  ℤ≥cuz 12240  ...cfz 12894  ..^cfzo 13037  seqcseq 13373  ♯chash 13695  Basecbs 16483  +gcplusg 16565  0gc0g 16713   Σg cgsu 16714  Mndcmnd 17911  Cntzccntz 18445 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-0g 16715  df-gsum 16716  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-cntz 18447 This theorem is referenced by:  gsumzadd  19042  dprdfadd  19142
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