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

Theorem gsumconst 19880
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 1191 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ 𝑋 ∈ 𝐡)
2 gsumconst.b . . . . . 6 𝐡 = (Baseβ€˜πΊ)
3 eqid 2727 . . . . . 6 (0gβ€˜πΊ) = (0gβ€˜πΊ)
4 gsumconst.m . . . . . 6 Β· = (.gβ€˜πΊ)
52, 3, 4mulg0 19021 . . . . 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 6891 . . . . . . 7 (𝐴 = βˆ… β†’ (β™―β€˜π΄) = (β™―β€˜βˆ…))
87adantl 481 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = (β™―β€˜βˆ…))
9 hash0 14350 . . . . . 6 (β™―β€˜βˆ…) = 0
108, 9eqtrdi 2783 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = 0)
1110oveq1d 7429 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ ((β™―β€˜π΄) Β· 𝑋) = (0 Β· 𝑋))
12 mpteq1 5235 . . . . . . . 8 (𝐴 = βˆ… β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ βˆ… ↦ 𝑋))
1312adantl 481 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ βˆ… ↦ 𝑋))
14 mpt0 6691 . . . . . . 7 (π‘˜ ∈ βˆ… ↦ 𝑋) = βˆ…
1513, 14eqtrdi 2783 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = βˆ…)
1615oveq2d 7430 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (𝐺 Ξ£g βˆ…))
173gsum0 18635 . . . . 5 (𝐺 Ξ£g βˆ…) = (0gβ€˜πΊ)
1816, 17eqtrdi 2783 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (0gβ€˜πΊ))
196, 11, 183eqtr4rd 2778 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋))
2019ex 412 . 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 12887 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
2321, 22eleqtrdi 2838 . . . . . . 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 484 . . . . . . . . 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 1191 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑋 ∈ 𝐡)
2625adantr 480 . . . . . . . . 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 2727 . . . . . . . . . 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 583 . . . . . . . 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 6833 . . . . . . . . . . . . 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 7088 . . . . . . . . . . 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 2728 . . . . . . . . . . 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 2728 . . . . . . . . . . 11 (π‘˜ = (π‘“β€˜π‘₯) β†’ 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 7132 . . . . . . . . . 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 6893 . . . . . . . . 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 480 . . . . . . . 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 13554 . . . . . . . . 9 (π‘₯ ∈ (1...(β™―β€˜π΄)) β†’ π‘₯ ∈ β„•)
40 fvconst2g 7208 . . . . . . . . 9 ((𝑋 ∈ 𝐡 ∧ π‘₯ ∈ β„•) β†’ ((β„• Γ— {𝑋})β€˜π‘₯) = 𝑋)
4125, 39, 40syl2an 595 . . . . . . . 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 2777 . . . . . . 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 14015 . . . . . 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 2727 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
45 eqid 2727 . . . . . . 7 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
46 simpl1 1189 . . . . . . 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 1190 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝐴 ∈ Fin)
4825adantr 480 . . . . . . . 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 7119 . . . . . . 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 2728 . . . . . . . . . 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 19267 . . . . . . . . . . 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 4808 . . . . . . . 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 4658 . . . . . . . . . . . 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 480 . . . . . . . . . 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 7119 . . . . . . . . 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 6724 . . . . . . . 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 19282 . . . . . . . 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 583 . . . . . . 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 6832 . . . . . . . 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 8175 . . . . . . . . 9 ((π‘˜ ∈ 𝐴 ↦ 𝑋) supp (0gβ€˜πΊ)) βŠ† dom (π‘˜ ∈ 𝐴 ↦ 𝑋)
65 eqid 2727 . . . . . . . . . . 11 (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ 𝐴 ↦ 𝑋)
6665dmmptss 6239 . . . . . . . . . 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 3989 . . . . . . . 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 6840 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))–onto→𝐴)
70 forn 6808 . . . . . . . . . 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 4019 . . . . . . 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 2727 . . . . . . 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 19853 . . . . . 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 2727 . . . . . . . 8 seq1((+gβ€˜πΊ), (β„• Γ— {𝑋})) = seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))
772, 44, 4, 76mulgnn 19022 . . . . . . 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 583 . . . . . 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 2777 . . . . 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 456 . . . 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 1929 . . 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 453 . 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 15680 . . 3 (𝐴 ∈ Fin β†’ (𝐴 = βˆ… ∨ ((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)))
84833ad2ant2 1132 . 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 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099   βŠ† wss 3944  βˆ…c0 4318  {csn 4624   ↦ cmpt 5225   Γ— cxp 5670  dom cdm 5672  ran crn 5673   ∘ ccom 5676  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   supp csupp 8159  Fincfn 8955  0cc0 11130  1c1 11131  β„•cn 12234  β„€β‰₯cuz 12844  ...cfz 13508  seqcseq 13990  β™―chash 14313  Basecbs 17171  +gcplusg 17224  0gc0g 17412   Ξ£g cgsu 17413  Mndcmnd 18685  .gcmg 19014  Cntzccntz 19257
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-seq 13991  df-hash 14314  df-0g 17414  df-gsum 17415  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-mulg 19015  df-cntz 19259
This theorem is referenced by:  gsumconstf  19881  mdetdiagid  22489  chpscmat  22731  chp0mat  22735  chpidmat  22736  tmdgsum2  23987  amgmlem  26909  lgseisenlem4  27298  gsumhashmul  32748
  Copyright terms: Public domain W3C validator