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Theorem gsumconst 19843
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 2730 . . . . . 6 (0gβ€˜πΊ) = (0gβ€˜πΊ)
4 gsumconst.m . . . . . 6 Β· = (.gβ€˜πΊ)
52, 3, 4mulg0 18993 . . . . 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 6890 . . . . . . 7 (𝐴 = βˆ… β†’ (β™―β€˜π΄) = (β™―β€˜βˆ…))
87adantl 480 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = (β™―β€˜βˆ…))
9 hash0 14331 . . . . . 6 (β™―β€˜βˆ…) = 0
108, 9eqtrdi 2786 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (β™―β€˜π΄) = 0)
1110oveq1d 7426 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ ((β™―β€˜π΄) Β· 𝑋) = (0 Β· 𝑋))
12 mpteq1 5240 . . . . . . . 8 (𝐴 = βˆ… β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = (π‘˜ ∈ βˆ… ↦ 𝑋))
1312adantl 480 . . . . . . 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 2786 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) = βˆ…)
1615oveq2d 7427 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (𝐺 Ξ£g βˆ…))
173gsum0 18609 . . . . 5 (𝐺 Ξ£g βˆ…) = (0gβ€˜πΊ)
1816, 17eqtrdi 2786 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = (0gβ€˜πΊ))
196, 11, 183eqtr4rd 2781 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) ∧ 𝐴 = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋))
2019ex 411 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐡) β†’ (𝐴 = βˆ… β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) = ((β™―β€˜π΄) Β· 𝑋)))
21 simprl 767 . . . . . . . 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 12869 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
2321, 22eleqtrdi 2841 . . . . . . 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 483 . . . . . . . . 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 479 . . . . . . . . 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 2730 . . . . . . . . . 10 (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋) = (π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)
2827fvmpt2 7008 . . . . . . . . 9 ((π‘₯ ∈ (1...(β™―β€˜π΄)) ∧ 𝑋 ∈ 𝐡) β†’ ((π‘₯ ∈ (1...(β™―β€˜π΄)) ↦ 𝑋)β€˜π‘₯) = 𝑋)
2924, 26, 28syl2anc 582 . . . . . . . 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 6832 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))⟢𝐴)
3130ad2antll 725 . . . . . . . . . . . 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 7085 . . . . . . . . . . 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 6959 . . . . . . . . . . 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 2731 . . . . . . . . . . 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 2731 . . . . . . . . . . 11 (π‘˜ = (π‘“β€˜π‘₯) β†’ 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 7128 . . . . . . . . . 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 6892 . . . . . . . . 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 479 . . . . . . . 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 13534 . . . . . . . . 9 (π‘₯ ∈ (1...(β™―β€˜π΄)) β†’ π‘₯ ∈ β„•)
40 fvconst2g 7204 . . . . . . . . 9 ((𝑋 ∈ 𝐡 ∧ π‘₯ ∈ β„•) β†’ ((β„• Γ— {𝑋})β€˜π‘₯) = 𝑋)
4125, 39, 40syl2an 594 . . . . . . . 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 2780 . . . . . . 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 13996 . . . . . 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 2730 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
45 eqid 2730 . . . . . . 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 479 . . . . . . . 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 2731 . . . . . . . . . 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 19230 . . . . . . . . . . 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 709 . . . . . . . . 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 4811 . . . . . . . 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 4661 . . . . . . . . . . . 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 479 . . . . . . . . . 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 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 19245 . . . . . . . 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 582 . . . . . . 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 6831 . . . . . . . 8 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))–1-1→𝐴)
6362ad2antll 725 . . . . . . 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 8164 . . . . . . . . 9 ((π‘˜ ∈ 𝐴 ↦ 𝑋) supp (0gβ€˜πΊ)) βŠ† dom (π‘˜ ∈ 𝐴 ↦ 𝑋)
65 eqid 2730 . . . . . . . . . . 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 3992 . . . . . . . 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 6839 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))–onto→𝐴)
70 forn 6807 . . . . . . . . . 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 725 . . . . . . . 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 4022 . . . . . . 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 2730 . . . . . . 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 19816 . . . . . 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 2730 . . . . . . . 8 seq1((+gβ€˜πΊ), (β„• Γ— {𝑋})) = seq1((+gβ€˜πΊ), (β„• Γ— {𝑋}))
772, 44, 4, 76mulgnn 18994 . . . . . . 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 582 . . . . . 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 2780 . . . . 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 455 . . . 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 1934 . . 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 452 . 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 15660 . . 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 856 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 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   βŠ† wss 3947  βˆ…c0 4321  {csn 4627   ↦ cmpt 5230   Γ— cxp 5673  dom cdm 5675  ran crn 5676   ∘ ccom 5679  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   supp csupp 8148  Fincfn 8941  0cc0 11112  1c1 11113  β„•cn 12216  β„€β‰₯cuz 12826  ...cfz 13488  seqcseq 13970  β™―chash 14294  Basecbs 17148  +gcplusg 17201  0gc0g 17389   Ξ£g cgsu 17390  Mndcmnd 18659  .gcmg 18986  Cntzccntz 19220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-0g 17391  df-gsum 17392  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mulg 18987  df-cntz 19222
This theorem is referenced by:  gsumconstf  19844  mdetdiagid  22322  chpscmat  22564  chp0mat  22568  chpidmat  22569  tmdgsum2  23820  amgmlem  26730  lgseisenlem4  27117  gsumhashmul  32478
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