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Theorem gsumconst 19802
Description: Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gsumconst.b 𝐡 = (Baseβ€˜πΊ)
gsumconst.m Β· = (.gβ€˜πΊ)
Assertion
Ref Expression
gsumconst ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋))
Distinct variable groups:   𝐴,π‘˜   𝐡,π‘˜   π‘˜,𝐺   π‘˜,𝑋
Allowed substitution hint:   Β· (π‘˜)

Proof of Theorem gsumconst
Dummy variables 𝑓 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1194 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ 𝑋 ∈ 𝐡)
2 gsumconst.b . . . . . 6 𝐡 = (Baseβ€˜πΊ)
3 eqid 2733 . . . . . 6 (0gβ€˜πΊ) = (0gβ€˜πΊ)
4 gsumconst.m . . . . . 6 Β· = (.gβ€˜πΊ)
52, 3, 4mulg0 18957 . . . . 5 (𝑋 ∈ 𝐡 β†’ (0 Β· 𝑋) = (0gβ€˜πΊ))
61, 5syl 17 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (0 Β· 𝑋) = (0gβ€˜πΊ))
7 fveq2 6892 . . . . . . 7 (𝐴 = βˆ… β†’ (β™―β€˜π΄) = (β™―β€˜βˆ…))
87adantl 483 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = (β™―β€˜βˆ…))
9 hash0 14327 . . . . . 6 (β™―β€˜βˆ…) = 0
108, 9eqtrdi 2789 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = 0)
1110oveq1d 7424 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ ((β™―β€˜π΄) Β· 𝑋) = (0 Β· 𝑋))
12 mpteq1 5242 . . . . . . . 8 (𝐴 = βˆ… β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ βˆ… ↦ 𝑋))
1312adantl 483 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ βˆ… ↦ 𝑋))
14 mpt0 6693 . . . . . . 7 (π‘˜ ∈ βˆ… ↦ 𝑋) = βˆ…
1513, 14eqtrdi 2789 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = βˆ…)
1615oveq2d 7425 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (𝐺 Ξ£g βˆ…))
173gsum0 18603 . . . . 5 (𝐺 Ξ£g βˆ…) = (0gβ€˜πΊ)
1816, 17eqtrdi 2789 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (0gβ€˜πΊ))
196, 11, 183eqtr4rd 2784 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋))
2019ex 414 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) β†’ (𝐴 = βˆ… β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋)))
21 simprl 770 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (β™―β€˜π΄) ∈ β„•)
22 nnuz 12865 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
2321, 22eleqtrdi 2844 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜1))
24 simpr 486 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ π‘₯ ∈ (1...(β™―β€˜π΄)))
25 simpl3 1194 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑋 ∈ 𝐡)
2625adantr 482 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ 𝑋 ∈ 𝐡)
27 eqid 2733 . . . . . . . . . 10 (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋) = (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)
2827fvmpt2 7010 . . . . . . . . 9 ((π‘₯ ∈ (1...(β™―β€˜π΄)) ∧ 𝑋 ∈ 𝐡) β†’ ((π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)β€˜π‘₯) = 𝑋)
2924, 26, 28syl2anc 585 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ ((π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)β€˜π‘₯) = 𝑋)
30 f1of 6834 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))⟢𝐴)
3130ad2antll 728 . . . . . . . . . . . 12 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑓:(1...(β™―β€˜π΄))⟢𝐴)
3231ffvelcdmda 7087 . . . . . . . . . . 11 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ (π‘“β€˜π‘₯) ∈ 𝐴)
3331feqmptd 6961 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑓 = (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ (π‘“β€˜π‘₯)))
34 eqidd 2734 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ 𝐴 ↦ 𝑋))
35 eqidd 2734 . . . . . . . . . . 11 (π‘˜ = (π‘“β€˜π‘₯) β†’ 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 7127 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) = (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋))
3736fveq1d 6894 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓)β€˜π‘₯) = ((π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)β€˜π‘₯))
3837adantr 482 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓)β€˜π‘₯) = ((π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)β€˜π‘₯))
39 elfznn 13530 . . . . . . . . 9 (π‘₯ ∈ (1...(β™―β€˜π΄)) β†’ π‘₯ ∈ β„•)
40 fvconst2g 7203 . . . . . . . . 9 ((𝑋 ∈ 𝐡 ∧ π‘₯ ∈ β„•) β†’ ((β„• Γ— {𝑋})β€˜π‘₯) = 𝑋)
4125, 39, 40syl2an 597 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ ((β„• Γ— {𝑋})β€˜π‘₯) = 𝑋)
4229, 38, 413eqtr4d 2783 . . . . . . 7 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘₯ ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓)β€˜π‘₯) = ((β„• Γ— {𝑋})β€˜π‘₯))
4323, 42seqfveq 13992 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (seq1((+gβ€˜πΊ), ((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓))β€˜(β™―β€˜π΄)) = (seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))β€˜(β™―β€˜π΄)))
44 eqid 2733 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
45 eqid 2733 . . . . . . 7 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
46 simpl1 1192 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝐺 ∈ Mnd)
47 simpl2 1193 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝐴 ∈ Fin)
4825adantr 482 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)
4948fmpttd 7115 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋):𝐴⟢𝐡)
50 eqidd 2734 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝑋(+gβ€˜πΊ)𝑋) = (𝑋(+gβ€˜πΊ)𝑋))
512, 44, 45elcntzsn 19189 . . . . . . . . . . 11 (𝑋 ∈ 𝐡 β†’ (𝑋 ∈ ((Cntzβ€˜πΊ)β€˜{𝑋}) ↔ (𝑋 ∈ 𝐡 ∧ (𝑋(+gβ€˜πΊ)𝑋) = (𝑋(+gβ€˜πΊ)𝑋))))
5225, 51syl 17 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝑋 ∈ ((Cntzβ€˜πΊ)β€˜{𝑋}) ↔ (𝑋 ∈ 𝐡 ∧ (𝑋(+gβ€˜πΊ)𝑋) = (𝑋(+gβ€˜πΊ)𝑋))))
5325, 50, 52mpbir2and 712 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑋 ∈ ((Cntzβ€˜πΊ)β€˜{𝑋}))
5453snssd 4813 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ {𝑋} βŠ† ((Cntzβ€˜πΊ)β€˜{𝑋}))
55 snidg 4663 . . . . . . . . . . . 12 (𝑋 ∈ 𝐡 β†’ 𝑋 ∈ {𝑋})
5625, 55syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑋 ∈ {𝑋})
5756adantr 482 . . . . . . . . . 10 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ {𝑋})
5857fmpttd 7115 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋):𝐴⟢{𝑋})
5958frnd 6726 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ran (π‘˜ ∈ 𝐴 ↦ 𝑋) βŠ† {𝑋})
6045cntzidss 19204 . . . . . . . 8 (({𝑋} βŠ† ((Cntzβ€˜πΊ)β€˜{𝑋}) ∧ ran (π‘˜ ∈ 𝐴 ↦ 𝑋) βŠ† {𝑋}) β†’ ran (π‘˜ ∈ 𝐴 ↦ 𝑋) βŠ† ((Cntzβ€˜πΊ)β€˜ran (π‘˜ ∈ 𝐴 ↦ 𝑋)))
6154, 59, 60syl2anc 585 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ran (π‘˜ ∈ 𝐴 ↦ 𝑋) βŠ† ((Cntzβ€˜πΊ)β€˜ran (π‘˜ ∈ 𝐴 ↦ 𝑋)))
62 f1of1 6833 . . . . . . . 8 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))–1-1→𝐴)
6362ad2antll 728 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑓:(1...(β™―β€˜π΄))–1-1→𝐴)
64 suppssdm 8162 . . . . . . . . 9 ((π‘˜ ∈ 𝐴 ↦ 𝑋) supp (0gβ€˜πΊ)) βŠ† dom (π‘˜ ∈ 𝐴 ↦ 𝑋)
65 eqid 2733 . . . . . . . . . . 11 (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ 𝐴 ↦ 𝑋)
6665dmmptss 6241 . . . . . . . . . 10 dom (π‘˜ ∈ 𝐴 ↦ 𝑋) βŠ† 𝐴
6766a1i 11 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ dom (π‘˜ ∈ 𝐴 ↦ 𝑋) βŠ† 𝐴)
6864, 67sstrid 3994 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝑋) supp (0gβ€˜πΊ)) βŠ† 𝐴)
69 f1ofo 6841 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))–onto→𝐴)
70 forn 6809 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΄))–onto→𝐴 β†’ ran 𝑓 = 𝐴)
7169, 70syl 17 . . . . . . . . 9 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ ran 𝑓 = 𝐴)
7271ad2antll 728 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ran 𝑓 = 𝐴)
7368, 72sseqtrrd 4024 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝑋) supp (0gβ€˜πΊ)) βŠ† ran 𝑓)
74 eqid 2733 . . . . . . 7 (((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) supp (0gβ€˜πΊ)) = (((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) supp (0gβ€˜πΊ))
752, 3, 44, 45, 46, 47, 49, 61, 21, 63, 73, 74gsumval3 19775 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (seq1((+gβ€˜πΊ), ((π‘˜ ∈ 𝐴 ↦ 𝑋) ∘ 𝑓))β€˜(β™―β€˜π΄)))
76 eqid 2733 . . . . . . . 8 seq1((+gβ€˜πΊ), (β„• Γ— {𝑋})) = seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))
772, 44, 4, 76mulgnn 18958 . . . . . . 7 (((β™―β€˜π΄) ∈ β„• ∧ 𝑋 ∈ 𝐡) β†’ ((β™―β€˜π΄) Β· 𝑋) = (seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))β€˜(β™―β€˜π΄)))
7821, 25, 77syl2anc 585 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ ((β™―β€˜π΄) Β· 𝑋) = (seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))β€˜(β™―β€˜π΄)))
7943, 75, 783eqtr4d 2783 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋))
8079expr 458 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ (β™―β€˜π΄) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋)))
8180exlimdv 1937 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ (β™―β€˜π΄) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋)))
8281expimpd 455 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) β†’ (((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋)))
83 fz1f1o 15656 . . 3 (𝐴 ∈ Fin β†’ (𝐴 = βˆ… ∨ ((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)))
84833ad2ant2 1135 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) β†’ (𝐴 = βˆ… ∨ ((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)))
8520, 82, 84mpjaod 859 1 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   βŠ† wss 3949  βˆ…c0 4323  {csn 4629   ↦ cmpt 5232   Γ— cxp 5675  dom cdm 5677  ran crn 5678   ∘ ccom 5681  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409   supp csupp 8146  Fincfn 8939  0cc0 11110  1c1 11111  β„•cn 12212  β„€β‰₯cuz 12822  ...cfz 13484  seqcseq 13966  β™―chash 14290  Basecbs 17144  +gcplusg 17197  0gc0g 17385   Ξ£g cgsu 17386  Mndcmnd 18625  .gcmg 18950  Cntzccntz 19179
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-0g 17387  df-gsum 17388  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mulg 18951  df-cntz 19181
This theorem is referenced by:  gsumconstf  19803  mdetdiagid  22102  chpscmat  22344  chp0mat  22348  chpidmat  22349  tmdgsum2  23600  amgmlem  26494  lgseisenlem4  26881  gsumhashmul  32208
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