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Theorem gsumconst 19718
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 2737 . . . . . 6 (0gβ€˜πΊ) = (0gβ€˜πΊ)
4 gsumconst.m . . . . . 6 Β· = (.gβ€˜πΊ)
52, 3, 4mulg0 18886 . . . . 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 6847 . . . . . . 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 14274 . . . . . 6 (β™―β€˜βˆ…) = 0
108, 9eqtrdi 2793 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = 0)
1110oveq1d 7377 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ ((β™―β€˜π΄) Β· 𝑋) = (0 Β· 𝑋))
12 mpteq1 5203 . . . . . . . 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 6648 . . . . . . 7 (π‘˜ ∈ βˆ… ↦ 𝑋) = βˆ…
1513, 14eqtrdi 2793 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = βˆ…)
1615oveq2d 7378 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (𝐺 Ξ£g βˆ…))
173gsum0 18546 . . . . 5 (𝐺 Ξ£g βˆ…) = (0gβ€˜πΊ)
1816, 17eqtrdi 2793 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (0gβ€˜πΊ))
196, 11, 183eqtr4rd 2788 . . 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 12813 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
2321, 22eleqtrdi 2848 . . . . . . 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 2737 . . . . . . . . . 10 (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋) = (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)
2827fvmpt2 6964 . . . . . . . . 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 6789 . . . . . . . . . . . . 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 7040 . . . . . . . . . . 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 6915 . . . . . . . . . . 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 2738 . . . . . . . . . . 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 2738 . . . . . . . . . . 11 (π‘˜ = (π‘“β€˜π‘₯) β†’ 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 7080 . . . . . . . . . 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 6849 . . . . . . . . 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 13477 . . . . . . . . 9 (π‘₯ ∈ (1...(β™―β€˜π΄)) β†’ π‘₯ ∈ β„•)
40 fvconst2g 7156 . . . . . . . . 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 2787 . . . . . . 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 13939 . . . . . 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 2737 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
45 eqid 2737 . . . . . . 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 7068 . . . . . . 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 2738 . . . . . . . . . 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 19112 . . . . . . . . . . 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 4774 . . . . . . . 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 4625 . . . . . . . . . . . 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 7068 . . . . . . . . 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 6681 . . . . . . . 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 19125 . . . . . . . 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 6788 . . . . . . . 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 8113 . . . . . . . . 9 ((π‘˜ ∈ 𝐴 ↦ 𝑋) supp (0gβ€˜πΊ)) βŠ† dom (π‘˜ ∈ 𝐴 ↦ 𝑋)
65 eqid 2737 . . . . . . . . . . 11 (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ 𝐴 ↦ 𝑋)
6665dmmptss 6198 . . . . . . . . . 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 3960 . . . . . . . 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 6796 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))–onto→𝐴)
70 forn 6764 . . . . . . . . . 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 3990 . . . . . . 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 2737 . . . . . . 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 19691 . . . . . 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 2737 . . . . . . . 8 seq1((+gβ€˜πΊ), (β„• Γ— {𝑋})) = seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))
772, 44, 4, 76mulgnn 18887 . . . . . . 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 2787 . . . . 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 15602 . . 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 3915  βˆ…c0 4287  {csn 4591   ↦ cmpt 5193   Γ— cxp 5636  dom cdm 5638  ran crn 5639   ∘ ccom 5642  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362   supp csupp 8097  Fincfn 8890  0cc0 11058  1c1 11059  β„•cn 12160  β„€β‰₯cuz 12770  ...cfz 13431  seqcseq 13913  β™―chash 14237  Basecbs 17090  +gcplusg 17140  0gc0g 17328   Ξ£g cgsu 17329  Mndcmnd 18563  .gcmg 18879  Cntzccntz 19102
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-seq 13914  df-hash 14238  df-0g 17330  df-gsum 17331  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mulg 18880  df-cntz 19104
This theorem is referenced by:  gsumconstf  19719  mdetdiagid  21965  chpscmat  22207  chp0mat  22211  chpidmat  22212  tmdgsum2  23463  amgmlem  26355  lgseisenlem4  26742  gsumhashmul  31940
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