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Theorem gsumzf1o 19774
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 18713 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
51, 2, 4syl2anc 584 . . . . . 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 6829 . . . . . . . . 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 7939 . . . . . . . 8 ((𝐻:𝐢–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐢 ∈ V)
108, 2, 9syl2anc 584 . . . . . . 7 (πœ‘ β†’ 𝐢 ∈ V)
113gsumz 18713 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐢 ∈ V) β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )) = 0 )
121, 10, 11syl2anc 584 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )) = 0 )
135, 12eqtr4d 2775 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )))
1413adantr 481 . . . 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 6902 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 (πœ‘ β†’ 0 ∈ V)
18 ssidd 4004 . . . . . 6 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐹 supp 0 ))
1915, 2, 17, 18gsumcllem 19770 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
2019oveq2d 7421 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
21 f1of 6830 . . . . . . . . 9 (𝐻:𝐢–1-1-onto→𝐴 β†’ 𝐻:𝐢⟢𝐴)
226, 21syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐻:𝐢⟢𝐴)
2322adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐻:𝐢⟢𝐴)
2423ffvelcdmda 7083 . . . . . 6 (((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) ∧ π‘₯ ∈ 𝐢) β†’ (π»β€˜π‘₯) ∈ 𝐴)
2523feqmptd 6957 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐻 = (π‘₯ ∈ 𝐢 ↦ (π»β€˜π‘₯)))
26 eqidd 2733 . . . . . 6 (π‘˜ = (π»β€˜π‘₯) β†’ 0 = 0 )
2724, 25, 19, 26fmptco 7123 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 ∘ 𝐻) = (π‘₯ ∈ 𝐢 ↦ 0 ))
2827oveq2d 7421 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 ∘ 𝐻)) = (𝐺 Ξ£g (π‘₯ ∈ 𝐢 ↦ 0 )))
2914, 20, 283eqtr4d 2782 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻)))
3029ex 413 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (𝐹 ∘ 𝐻))))
316adantr 481 . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . 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 5859 . . . . . . . . . . . 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 6829 . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 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 8158 . . . . . . . . . . . . . . . 16 (𝐹 supp 0 ) βŠ† dom 𝐹
3837, 15fssdm 6734 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
3938adantr 481 . . . . . . . . . . . . . 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 6790 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 6784 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴 β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))⟢𝐴)
43 fcoi2 6763 . . . . . . . . . . . . 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 2772 . . . . . . . . . . 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 6261 . . . . . . . . . . 11 ((𝐻 ∘ ◑𝐻) ∘ 𝑓) = (𝐻 ∘ (◑𝐻 ∘ 𝑓))
4745, 46eqtr3di 2787 . . . . . . . . . 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 5860 . . . . . . . . 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 6261 . . . . . . . . 9 ((𝐹 ∘ 𝐻) ∘ (◑𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◑𝐻 ∘ 𝑓)))
5048, 49eqtr4di 2790 . . . . . . . 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 13970 . . . . . . 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 6890 . . . . . 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 2732 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
55 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
561adantr 481 . . . . . . 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 481 . . . . . . 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 481 . . . . . . 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 481 . . . . . . 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 769 . . . . . . 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 4003 . . . . . . . 8 (𝐹 supp 0 ) βŠ† (𝐹 supp 0 )
63 f1ofo 6837 . . . . . . . . . 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 6805 . . . . . . . . . 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 727 . . . . . . . 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 4034 . . . . . . 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 2732 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
6953, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68gsumval3 19769 . . . . . 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 481 . . . . . . 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 6738 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐻:𝐢⟢𝐴) β†’ (𝐹 ∘ 𝐻):𝐢⟢𝐡)
7215, 22, 71syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (𝐹 ∘ 𝐻):𝐢⟢𝐡)
7372adantr 481 . . . . . . 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 5969 . . . . . . . . 9 ran (𝐹 ∘ 𝐻) βŠ† ran 𝐹
7555cntzidss 19198 . . . . . . . . 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 586 . . . . . . . 8 (πœ‘ β†’ ran (𝐹 ∘ 𝐻) βŠ† (π‘β€˜ran (𝐹 ∘ 𝐻)))
7776adantr 481 . . . . . . 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 6842 . . . . . . . . . 10 (𝐻:𝐢–1-1-onto→𝐴 β†’ ◑𝐻:𝐴–1-1-onto→𝐢)
79 f1of1 6829 . . . . . . . . . 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 481 . . . . . . . 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 6796 . . . . . . . 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 584 . . . . . . 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 4004 . . . . . . . . . . 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 7225 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐹 ∈ V)
86 suppimacnv 8155 . . . . . . . . . . . . . 14 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
8785, 16, 86sylancl 586 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝐹 supp 0 ) = (◑𝐹 β€œ (V βˆ– { 0 })))
8887eqcomd 2738 . . . . . . . . . . . 12 (πœ‘ β†’ (◑𝐹 β€œ (V βˆ– { 0 })) = (𝐹 supp 0 ))
8988adantr 481 . . . . . . . . . . 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 4028 . . . . . . . . . 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 6098 . . . . . . . . . 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 5883 . . . . . . . . . . 11 β—‘(𝐹 ∘ 𝐻) = (◑𝐻 ∘ ◑𝐹)
9493imaeq1i 6054 . . . . . . . . . 10 (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) = ((◑𝐻 ∘ ◑𝐹) β€œ (V βˆ– { 0 }))
95 imaco 6247 . . . . . . . . . 10 ((◑𝐻 ∘ ◑𝐹) β€œ (V βˆ– { 0 })) = (◑𝐻 β€œ (◑𝐹 β€œ (V βˆ– { 0 })))
9694, 95eqtri 2760 . . . . . . . . 9 (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })) = (◑𝐻 β€œ (◑𝐹 β€œ (V βˆ– { 0 })))
97 rnco2 6249 . . . . . . . . 9 ran (◑𝐻 ∘ 𝑓) = (◑𝐻 β€œ ran 𝑓)
9892, 96, 973sstr4g 4026 . . . . . . . 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 7914 . . . . . . . . . . . . 13 ((𝐻:𝐢–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐻 ∈ V)
1006, 2, 99syl2anc 584 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐻 ∈ V)
101 coexg 7916 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ (𝐹 ∘ 𝐻) ∈ V)
10285, 100, 101syl2anc 584 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹 ∘ 𝐻) ∈ V)
103 suppimacnv 8155 . . . . . . . . . . 11 (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) β†’ ((𝐹 ∘ 𝐻) supp 0 ) = (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })))
104102, 16, 103sylancl 586 . . . . . . . . . 10 (πœ‘ β†’ ((𝐹 ∘ 𝐻) supp 0 ) = (β—‘(𝐹 ∘ 𝐻) β€œ (V βˆ– { 0 })))
105104sseq1d 4012 . . . . . . . . 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 481 . . . . . . . 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 256 . . . . . . 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 2732 . . . . . . 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 19769 . . . . . 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 2782 . . . . 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 457 . . . 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 1936 . . 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 454 . 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 9364 . . . 4 (𝐹 finSupp 0 β†’ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
116115simprd 496 . . 3 (𝐹 finSupp 0 β†’ (𝐹 supp 0 ) ∈ Fin)
117 fz1f1o 15652 . . 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 858 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 396   ∨ wo 845   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  {csn 4627   class class class wbr 5147   ↦ cmpt 5230   I cid 5572  β—‘ccnv 5674  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   ∘ ccom 5679  Fun wfun 6534  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   supp csupp 8142  Fincfn 8935   finSupp cfsupp 9357  1c1 11107  β„•cn 12208  ...cfz 13480  seqcseq 13962  β™―chash 14286  Basecbs 17140  +gcplusg 17193  0gc0g 17381   Ξ£g cgsu 17382  Mndcmnd 18621  Cntzccntz 19173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-0g 17383  df-gsum 17384  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-cntz 19175
This theorem is referenced by:  gsumf1o  19778  smadiadetlem3  22161
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