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Theorem gsumzf1o 19696
Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 2-Jun-2019.)
Hypotheses
Ref Expression
gsumzcl.b 𝐡 = (Baseβ€˜πΊ)
gsumzcl.0 0 = (0gβ€˜πΊ)
gsumzcl.z 𝑍 = (Cntzβ€˜πΊ)
gsumzcl.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumzcl.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumzcl.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumzcl.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumzcl.w (πœ‘ β†’ 𝐹 finSupp 0 )
gsumzf1o.h (πœ‘ β†’ 𝐻:𝐢–1-1-onto→𝐴)
Assertion
Ref Expression
gsumzf1o (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻)))

Proof of Theorem gsumzf1o
Dummy variables 𝑓 π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ Mnd)
2 gsumzcl.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ 𝑉)
3 gsumzcl.0 . . . . . . . 8 0 = (0gβ€˜πΊ)
43gsumz 18653 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
51, 2, 4syl2anc 585 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
6 gsumzf1o.h . . . . . . . . 9 (πœ‘ β†’ 𝐻:𝐢–1-1-onto→𝐴)
7 f1of1 6788 . . . . . . . . 9 (𝐻:𝐢–1-1-onto→𝐴 β†’ 𝐻:𝐢–1-1→𝐴)
86, 7syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐻:𝐢–1-1→𝐴)
9 f1dmex 7894 . . . . . . . 8 ((𝐻:𝐢–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐢 ∈ V)
108, 2, 9syl2anc 585 . . . . . . 7 (πœ‘ β†’ 𝐢 ∈ V)
113gsumz 18653 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐢 ∈ V) β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )) = 0 )
121, 10, 11syl2anc 585 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )) = 0 )
135, 12eqtr4d 2780 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )))
1413adantr 482 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )))
15 gsumzcl.f . . . . . 6 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
163fvexi 6861 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 (πœ‘ β†’ 0 ∈ V)
18 ssidd 3972 . . . . . 6 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐹 supp 0 ))
1915, 2, 17, 18gsumcllem 19692 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
2019oveq2d 7378 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
21 f1of 6789 . . . . . . . . 9 (𝐻:𝐢–1-1-onto→𝐴 β†’ 𝐻:𝐢⟢𝐴)
226, 21syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐻:𝐢⟢𝐴)
2322adantr 482 . . . . . . 7 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐻:𝐢⟢𝐴)
2423ffvelcdmda 7040 . . . . . 6 (((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) ∧ π‘₯ ∈ 𝐢) β†’ (π»β€˜π‘₯) ∈ 𝐴)
2523feqmptd 6915 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐻 = (π‘₯ ∈ 𝐢 ↦ (π»β€˜π‘₯)))
26 eqidd 2738 . . . . . 6 (π‘˜ = (π»β€˜π‘₯) β†’ 0 = 0 )
2724, 25, 19, 26fmptco 7080 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 ∘ 𝐻) = (π‘₯ ∈ 𝐢 ↦ 0 ))
2827oveq2d 7378 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 ∘ 𝐻)) = (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )))
2914, 20, 283eqtr4d 2787 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻)))
3029ex 414 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻))))
316adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐻:𝐢–1-1-onto→𝐴)
32 f1ococnv2 6816 . . . . . . . . . . . . . 14 (𝐻:𝐢–1-1-onto→𝐴 β†’ (𝐻 ∘ ◑𝐻) = ( I β†Ύ 𝐴))
3331, 32syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐻 ∘ ◑𝐻) = ( I β†Ύ 𝐴))
3433coeq1d 5822 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ((𝐻 ∘ ◑𝐻) ∘ 𝑓) = (( I β†Ύ 𝐴) ∘ 𝑓))
35 f1of1 6788 . . . . . . . . . . . . . . 15 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ))
3635ad2antll 728 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ))
37 suppssdm 8113 . . . . . . . . . . . . . . . 16 (𝐹 supp 0 ) βŠ† dom 𝐹
3837, 15fssdm 6693 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
3938adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† 𝐴)
40 f1ss 6749 . . . . . . . . . . . . . 14 ((𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) βŠ† 𝐴) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴)
4136, 39, 40syl2anc 585 . . . . . . . . . . . . 13 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴)
42 f1f 6743 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴 β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐴)
43 fcoi2 6722 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐴 β†’ (( I β†Ύ 𝐴) ∘ 𝑓) = 𝑓)
4441, 42, 433syl 18 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (( I β†Ύ 𝐴) ∘ 𝑓) = 𝑓)
4534, 44eqtrd 2777 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ((𝐻 ∘ ◑𝐻) ∘ 𝑓) = 𝑓)
46 coass 6222 . . . . . . . . . . 11 ((𝐻 ∘ ◑𝐻) ∘ 𝑓) = (𝐻 ∘ (◑𝐻 ∘ 𝑓))
4745, 46eqtr3di 2792 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓 = (𝐻 ∘ (◑𝐻 ∘ 𝑓)))
4847coeq2d 5823 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝐻 ∘ (◑𝐻 ∘ 𝑓))))
49 coass 6222 . . . . . . . . 9 ((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◑𝐻 ∘ 𝑓)))
5048, 49eqtr4di 2795 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 ∘ 𝑓) = ((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)))
5150seqeq3d 13921 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓)) = seq1((+gβ€˜πΊ), ((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓))))
5251fveq1d 6849 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))) = (seq1((+gβ€˜πΊ), ((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 ))))
53 gsumzcl.b . . . . . . 7 𝐡 = (Baseβ€˜πΊ)
54 eqid 2737 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
55 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
561adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐺 ∈ Mnd)
572adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐴 ∈ 𝑉)
5815adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐹:𝐴⟢𝐡)
59 gsumzcl.c . . . . . . . 8 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
6059adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
61 simprl 770 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (β™―β€˜(𝐹 supp 0 )) ∈ β„•)
62 ssid 3971 . . . . . . . 8 (𝐹 supp 0 ) βŠ† (𝐹 supp 0 )
63 f1ofo 6796 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–ontoβ†’(𝐹 supp 0 ))
64 forn 6764 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–ontoβ†’(𝐹 supp 0 ) β†’ ran 𝑓 = (𝐹 supp 0 ))
6563, 64syl 17 . . . . . . . . 9 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ ran 𝑓 = (𝐹 supp 0 ))
6665ad2antll 728 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝑓 = (𝐹 supp 0 ))
6762, 66sseqtrrid 4002 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† ran 𝑓)
68 eqid 2737 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
6953, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68gsumval3 19691 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g 𝐹) = (seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))))
7010adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐢 ∈ V)
71 fco 6697 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐻:𝐢⟢𝐴) β†’ (𝐹 ∘ 𝐻):𝐢⟢𝐡)
7215, 22, 71syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (𝐹 ∘ 𝐻):𝐢⟢𝐡)
7372adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 ∘ 𝐻):𝐢⟢𝐡)
74 rncoss 5932 . . . . . . . . 9 ran (𝐹 ∘ 𝐻) βŠ† ran 𝐹
7555cntzidss 19125 . . . . . . . . 9 ((ran 𝐹 βŠ† (π‘β€˜ran 𝐹) ∧ ran (𝐹 ∘ 𝐻) βŠ† ran 𝐹) β†’ ran (𝐹 ∘ 𝐻) βŠ† (π‘β€˜ran (𝐹 ∘ 𝐻)))
7659, 74, 75sylancl 587 . . . . . . . 8 (πœ‘ β†’ ran (𝐹 ∘ 𝐻) βŠ† (π‘β€˜ran (𝐹 ∘ 𝐻)))
7776adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran (𝐹 ∘ 𝐻) βŠ† (π‘β€˜ran (𝐹 ∘ 𝐻)))
78 f1ocnv 6801 . . . . . . . . . 10 (𝐻:𝐢–1-1-onto→𝐴 β†’ ◑𝐻:𝐴–1-1-onto→𝐢)
79 f1of1 6788 . . . . . . . . . 10 (◑𝐻:𝐴–1-1-onto→𝐢 β†’ ◑𝐻:𝐴–1-1→𝐢)
806, 78, 793syl 18 . . . . . . . . 9 (πœ‘ β†’ ◑𝐻:𝐴–1-1→𝐢)
8180adantr 482 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ◑𝐻:𝐴–1-1→𝐢)
82 f1co 6755 . . . . . . . 8 ((◑𝐻:𝐴–1-1→𝐢 ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴) β†’ (◑𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐢)
8381, 41, 82syl2anc 585 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (◑𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐢)
84 ssidd 3972 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† (𝐹 supp 0 ))
8515, 2fexd 7182 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐹 ∈ V)
86 suppimacnv 8110 . . . . . . . . . . . . . 14 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
8785, 16, 86sylancl 587 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
8887eqcomd 2743 . . . . . . . . . . . 12 (πœ‘ β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = (𝐹 supp 0 ))
8988adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = (𝐹 supp 0 ))
9084, 89, 663sstr4d 3996 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (◑𝐹 β€œ (V βˆ– { 0 })) βŠ† ran 𝑓)
91 imass2 6059 . . . . . . . . . 10 ((◑𝐹 β€œ (V βˆ– { 0 })) βŠ† ran 𝑓 β†’ (◑𝐻 β€œ (◑𝐹 β€œ (V βˆ– { 0 }))) βŠ† (◑𝐻 β€œ ran 𝑓))
9290, 91syl 17 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (◑𝐻 β€œ (◑𝐹 β€œ (V βˆ– { 0 }))) βŠ† (◑𝐻 β€œ ran 𝑓))
93 cnvco 5846 . . . . . . . . . . 11 β—‘(𝐹 ∘ 𝐻) = (◑𝐻 ∘ ◑𝐹)
9493imaeq1i 6015 . . . . . . . . . 10 (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) = ((◑𝐻 ∘ ◑𝐹) β€œ (V βˆ– { 0 }))
95 imaco 6208 . . . . . . . . . 10 ((◑𝐻 ∘ ◑𝐹) β€œ (V βˆ– { 0 })) = (◑𝐻 β€œ (◑𝐹 β€œ (V βˆ– { 0 })))
9694, 95eqtri 2765 . . . . . . . . 9 (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) = (◑𝐻 β€œ (◑𝐹 β€œ (V βˆ– { 0 })))
97 rnco2 6210 . . . . . . . . 9 ran (◑𝐻 ∘ 𝑓) = (◑𝐻 β€œ ran 𝑓)
9892, 96, 973sstr4g 3994 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) βŠ† ran (◑𝐻 ∘ 𝑓))
99 f1oexrnex 7869 . . . . . . . . . . . . 13 ((𝐻:𝐢–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐻 ∈ V)
1006, 2, 99syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐻 ∈ V)
101 coexg 7871 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ (𝐹 ∘ 𝐻) ∈ V)
10285, 100, 101syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹 ∘ 𝐻) ∈ V)
103 suppimacnv 8110 . . . . . . . . . . 11 (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) β†’ ((𝐹 ∘ 𝐻) supp 0 ) = (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })))
104102, 16, 103sylancl 587 . . . . . . . . . 10 (πœ‘ β†’ ((𝐹 ∘ 𝐻) supp 0 ) = (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })))
105104sseq1d 3980 . . . . . . . . 9 (πœ‘ β†’ (((𝐹 ∘ 𝐻) supp 0 ) βŠ† ran (◑𝐻 ∘ 𝑓) ↔ (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) βŠ† ran (◑𝐻 ∘ 𝑓)))
106105adantr 482 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (((𝐹 ∘ 𝐻) supp 0 ) βŠ† ran (◑𝐻 ∘ 𝑓) ↔ (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) βŠ† ran (◑𝐻 ∘ 𝑓)))
10798, 106mpbird 257 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ((𝐹 ∘ 𝐻) supp 0 ) βŠ† ran (◑𝐻 ∘ 𝑓))
108 eqid 2737 . . . . . . 7 (((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)) supp 0 ) = (((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)) supp 0 )
10953, 3, 54, 55, 56, 70, 73, 77, 61, 83, 107, 108gsumval3 19691 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g (𝐹 ∘ 𝐻)) = (seq1((+gβ€˜πΊ), ((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 ))))
11052, 69, 1093eqtr4d 2787 . . . . 5 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻)))
111110expr 458 . . . 4 ((πœ‘ ∧ (β™―β€˜(𝐹 supp 0 )) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻))))
112111exlimdv 1937 . . 3 ((πœ‘ ∧ (β™―β€˜(𝐹 supp 0 )) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻))))
113112expimpd 455 . 2 (πœ‘ β†’ (((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻))))
114 gsumzcl.w . . 3 (πœ‘ β†’ 𝐹 finSupp 0 )
115 fsuppimp 9318 . . . 4 (𝐹 finSupp 0 β†’ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
116115simprd 497 . . 3 (𝐹 finSupp 0 β†’ (𝐹 supp 0 ) ∈ Fin)
117 fz1f1o 15602 . . 3 ((𝐹 supp 0 ) ∈ Fin β†’ ((𝐹 supp 0 ) = βˆ… ∨ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))))
118114, 116, 1173syl 18 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… ∨ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))))
11930, 113, 118mpjaod 859 1 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915  βˆ…c0 4287  {csn 4591   class class class wbr 5110   ↦ cmpt 5193   I cid 5535  β—‘ccnv 5637  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641   ∘ ccom 5642  Fun wfun 6495  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   finSupp cfsupp 9312  1c1 11059  β„•cn 12160  ...cfz 13431  seqcseq 13913  β™―chash 14237  Basecbs 17090  +gcplusg 17140  0gc0g 17328   Ξ£g cgsu 17329  Mndcmnd 18563  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-fsupp 9313  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-cntz 19104
This theorem is referenced by:  gsumf1o  19700  smadiadetlem3  22033
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