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Theorem gsumval3lem2 19774
Description: Lemma 2 for gsumval3 19775. (Contributed by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumval3.b 𝐡 = (Baseβ€˜πΊ)
gsumval3.0 0 = (0gβ€˜πΊ)
gsumval3.p + = (+gβ€˜πΊ)
gsumval3.z 𝑍 = (Cntzβ€˜πΊ)
gsumval3.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumval3.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumval3.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumval3.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumval3.m (πœ‘ β†’ 𝑀 ∈ β„•)
gsumval3.h (πœ‘ β†’ 𝐻:(1...𝑀)–1-1→𝐴)
gsumval3.n (πœ‘ β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
gsumval3.w π‘Š = ((𝐹 ∘ 𝐻) supp 0 )
Assertion
Ref Expression
gsumval3lem2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐺 Ξ£g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)))
Distinct variable groups:   + ,𝑓   𝐴,𝑓   πœ‘,𝑓   𝑓,𝐺   𝑓,𝑀   𝐡,𝑓   𝑓,𝐹   𝑓,𝐻   𝑓,π‘Š
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(𝑓)

Proof of Theorem gsumval3lem2
Dummy variables 𝑔 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3.h . . . . . . 7 (πœ‘ β†’ 𝐻:(1...𝑀)–1-1→𝐴)
2 f1f 6788 . . . . . . 7 (𝐻:(1...𝑀)–1-1→𝐴 β†’ 𝐻:(1...𝑀)⟢𝐴)
31, 2syl 17 . . . . . 6 (πœ‘ β†’ 𝐻:(1...𝑀)⟢𝐴)
4 fzfid 13938 . . . . . 6 (πœ‘ β†’ (1...𝑀) ∈ Fin)
53, 4fexd 7229 . . . . 5 (πœ‘ β†’ 𝐻 ∈ V)
6 vex 3479 . . . . 5 𝑓 ∈ V
7 coexg 7920 . . . . 5 ((𝐻 ∈ V ∧ 𝑓 ∈ V) β†’ (𝐻 ∘ 𝑓) ∈ V)
85, 6, 7sylancl 587 . . . 4 (πœ‘ β†’ (𝐻 ∘ 𝑓) ∈ V)
98ad2antrr 725 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓) ∈ V)
10 gsumval3.b . . . . 5 𝐡 = (Baseβ€˜πΊ)
11 gsumval3.0 . . . . 5 0 = (0gβ€˜πΊ)
12 gsumval3.p . . . . 5 + = (+gβ€˜πΊ)
13 gsumval3.z . . . . 5 𝑍 = (Cntzβ€˜πΊ)
14 gsumval3.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ Mnd)
15 gsumval3.a . . . . 5 (πœ‘ β†’ 𝐴 ∈ 𝑉)
16 gsumval3.f . . . . 5 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
17 gsumval3.c . . . . 5 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
18 gsumval3.m . . . . 5 (πœ‘ β†’ 𝑀 ∈ β„•)
19 gsumval3.n . . . . 5 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
20 gsumval3.w . . . . 5 π‘Š = ((𝐹 ∘ 𝐻) supp 0 )
2110, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 20gsumval3lem1 19773 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))
22 fzfi 13937 . . . . . . . 8 (1...𝑀) ∈ Fin
23 suppssdm 8162 . . . . . . . . . 10 ((𝐹 ∘ 𝐻) supp 0 ) βŠ† dom (𝐹 ∘ 𝐻)
2420, 23eqsstri 4017 . . . . . . . . 9 π‘Š βŠ† dom (𝐹 ∘ 𝐻)
2516, 3fcod 6744 . . . . . . . . 9 (πœ‘ β†’ (𝐹 ∘ 𝐻):(1...𝑀)⟢𝐡)
2624, 25fssdm 6738 . . . . . . . 8 (πœ‘ β†’ π‘Š βŠ† (1...𝑀))
27 ssfi 9173 . . . . . . . 8 (((1...𝑀) ∈ Fin ∧ π‘Š βŠ† (1...𝑀)) β†’ π‘Š ∈ Fin)
2822, 26, 27sylancr 588 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Fin)
2928ad2antrr 725 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š ∈ Fin)
301ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐻:(1...𝑀)–1-1→𝐴)
3126ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š βŠ† (1...𝑀))
32 f1ores 6848 . . . . . . . 8 ((𝐻:(1...𝑀)–1-1→𝐴 ∧ π‘Š βŠ† (1...𝑀)) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š))
3330, 31, 32syl2anc 585 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š))
3420imaeq2i 6058 . . . . . . . . . 10 (𝐻 β€œ π‘Š) = (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 ))
3516, 15fexd 7229 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹 ∈ V)
36 ovex 7442 . . . . . . . . . . . . . 14 (1...𝑀) ∈ V
37 fex 7228 . . . . . . . . . . . . . 14 ((𝐻:(1...𝑀)⟢𝐴 ∧ (1...𝑀) ∈ V) β†’ 𝐻 ∈ V)
383, 36, 37sylancl 587 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐻 ∈ V)
3935, 38jca 513 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐹 ∈ V ∧ 𝐻 ∈ V))
40 f1fun 6790 . . . . . . . . . . . . . 14 (𝐻:(1...𝑀)–1-1→𝐴 β†’ Fun 𝐻)
411, 40syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ Fun 𝐻)
4241, 19jca 513 . . . . . . . . . . . 12 (πœ‘ β†’ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻))
43 imacosupp 8194 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ ((Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
4439, 42, 43sylc 65 . . . . . . . . . . 11 (πœ‘ β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
4544adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘Š β‰  βˆ…) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
4634, 45eqtrid 2785 . . . . . . . . 9 ((πœ‘ ∧ π‘Š β‰  βˆ…) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
4746adantr 482 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
4847f1oeq3d 6831 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š) ↔ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 )))
4933, 48mpbid 231 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ))
5029, 49hasheqf1od 14313 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (β™―β€˜π‘Š) = (β™―β€˜(𝐹 supp 0 )))
5150fveq2d 6896 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 ))))
5221, 51jca 513 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 )))))
53 f1oeq1 6822 . . . 4 (𝑔 = (𝐻 ∘ 𝑓) β†’ (𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )))
54 coeq2 5859 . . . . . . 7 (𝑔 = (𝐻 ∘ 𝑓) β†’ (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
5554seqeq3d 13974 . . . . . 6 (𝑔 = (𝐻 ∘ 𝑓) β†’ seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
5655fveq1d 6894 . . . . 5 (𝑔 = (𝐻 ∘ 𝑓) β†’ (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 ))))
5756eqeq2d 2744 . . . 4 (𝑔 = (𝐻 ∘ 𝑓) β†’ ((seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))) ↔ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 )))))
5853, 57anbi12d 632 . . 3 (𝑔 = (𝐻 ∘ 𝑓) β†’ ((𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ ((𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜(𝐹 supp 0 ))))))
599, 52, 58spcedv 3589 . 2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))))
6014ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐺 ∈ Mnd)
6115ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐴 ∈ 𝑉)
6216ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐹:𝐴⟢𝐡)
6317ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
64 f1f1orn 6845 . . . . . . . . . . . . 13 (𝐻:(1...𝑀)–1-1→𝐴 β†’ 𝐻:(1...𝑀)–1-1-ontoβ†’ran 𝐻)
651, 64syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐻:(1...𝑀)–1-1-ontoβ†’ran 𝐻)
66 f1oen3g 8962 . . . . . . . . . . . 12 ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-ontoβ†’ran 𝐻) β†’ (1...𝑀) β‰ˆ ran 𝐻)
675, 65, 66syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ (1...𝑀) β‰ˆ ran 𝐻)
68 enfi 9190 . . . . . . . . . . 11 ((1...𝑀) β‰ˆ ran 𝐻 β†’ ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
6967, 68syl 17 . . . . . . . . . 10 (πœ‘ β†’ ((1...𝑀) ∈ Fin ↔ ran 𝐻 ∈ Fin))
7022, 69mpbii 232 . . . . . . . . 9 (πœ‘ β†’ ran 𝐻 ∈ Fin)
7170, 19ssfid 9267 . . . . . . . 8 (πœ‘ β†’ (𝐹 supp 0 ) ∈ Fin)
7271ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐹 supp 0 ) ∈ Fin)
7320neeq1i 3006 . . . . . . . . . 10 (π‘Š β‰  βˆ… ↔ ((𝐹 ∘ 𝐻) supp 0 ) β‰  βˆ…)
74 supp0cosupp0 8193 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ ((𝐹 supp 0 ) = βˆ… β†’ ((𝐹 ∘ 𝐻) supp 0 ) = βˆ…))
7574necon3d 2962 . . . . . . . . . . 11 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ (((𝐹 ∘ 𝐻) supp 0 ) β‰  βˆ… β†’ (𝐹 supp 0 ) β‰  βˆ…))
7635, 38, 75syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (((𝐹 ∘ 𝐻) supp 0 ) β‰  βˆ… β†’ (𝐹 supp 0 ) β‰  βˆ…))
7773, 76biimtrid 241 . . . . . . . . 9 (πœ‘ β†’ (π‘Š β‰  βˆ… β†’ (𝐹 supp 0 ) β‰  βˆ…))
7877imp 408 . . . . . . . 8 ((πœ‘ ∧ π‘Š β‰  βˆ…) β†’ (𝐹 supp 0 ) β‰  βˆ…)
7978adantr 482 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐹 supp 0 ) β‰  βˆ…)
8019ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
813frnd 6726 . . . . . . . . 9 (πœ‘ β†’ ran 𝐻 βŠ† 𝐴)
8281ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ran 𝐻 βŠ† 𝐴)
8380, 82sstrd 3993 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐹 supp 0 ) βŠ† 𝐴)
8410, 11, 12, 13, 60, 61, 62, 63, 72, 79, 83gsumval3eu 19772 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ βˆƒ!π‘₯βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))))
85 iota1 6521 . . . . . 6 (βˆƒ!π‘₯βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) β†’ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (β„©π‘₯βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))))) = π‘₯))
8684, 85syl 17 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (β„©π‘₯βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))))) = π‘₯))
87 eqid 2733 . . . . . . 7 (𝐹 supp 0 ) = (𝐹 supp 0 )
88 simprl 770 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ Β¬ 𝐴 ∈ ran ...)
8910, 11, 12, 13, 60, 61, 62, 63, 72, 79, 87, 88gsumval3a 19771 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐺 Ξ£g 𝐹) = (β„©π‘₯βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))))))
9089eqeq1d 2735 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐺 Ξ£g 𝐹) = π‘₯ ↔ (β„©π‘₯βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))))) = π‘₯))
9186, 90bitr4d 282 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = π‘₯))
9291alrimiv 1931 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ βˆ€π‘₯(βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = π‘₯))
93 fvex 6905 . . . 4 (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) ∈ V
94 eqeq1 2737 . . . . . . 7 (π‘₯ = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) β†’ (π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))) ↔ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))))
9594anbi2d 630 . . . . . 6 (π‘₯ = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) β†’ ((𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))))))
9695exbidv 1925 . . . . 5 (π‘₯ = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) β†’ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 ))))))
97 eqeq2 2745 . . . . 5 (π‘₯ = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) β†’ ((𝐺 Ξ£g 𝐹) = π‘₯ ↔ (𝐺 Ξ£g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š))))
9896, 97bibi12d 346 . . . 4 (π‘₯ = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) β†’ ((βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = π‘₯) ↔ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)))))
9993, 98spcv 3596 . . 3 (βˆ€π‘₯(βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = π‘₯) β†’ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š))))
10092, 99syl 17 . 2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (βˆƒπ‘”(𝑔:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) ∧ (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜(𝐹 supp 0 )))) ↔ (𝐺 Ξ£g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š))))
10159, 100mpbid 231 1 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐺 Ξ£g 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒ!weu 2563   β‰  wne 2941  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323   class class class wbr 5149  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680   ∘ ccom 5681  β„©cio 6494  Fun wfun 6538  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544   Isom wiso 6545  (class class class)co 7409   supp csupp 8146   β‰ˆ cen 8936  Fincfn 8939  1c1 11111   < clt 11248  β„•cn 12212  ...cfz 13484  seqcseq 13966  β™―chash 14290  Basecbs 17144  +gcplusg 17197  0gc0g 17385   Ξ£g cgsu 17386  Mndcmnd 18625  Cntzccntz 19179
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-0g 17387  df-gsum 17388  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-cntz 19181
This theorem is referenced by:  gsumval3  19775
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