MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumval3lem2 Structured version   Visualization version   GIF version

Theorem gsumval3lem2 19768
Description: Lemma 2 for gsumval3 19769. (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 6784 . . . . . . 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 13934 . . . . . 6 (πœ‘ β†’ (1...𝑀) ∈ Fin)
53, 4fexd 7225 . . . . 5 (πœ‘ β†’ 𝐻 ∈ V)
6 vex 3478 . . . . 5 𝑓 ∈ V
7 coexg 7916 . . . . 5 ((𝐻 ∈ V ∧ 𝑓 ∈ V) β†’ (𝐻 ∘ 𝑓) ∈ V)
85, 6, 7sylancl 586 . . . 4 (πœ‘ β†’ (𝐻 ∘ 𝑓) ∈ V)
98ad2antrr 724 . . 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 19767 . . . 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 13933 . . . . . . . 8 (1...𝑀) ∈ Fin
23 suppssdm 8158 . . . . . . . . . 10 ((𝐹 ∘ 𝐻) supp 0 ) βŠ† dom (𝐹 ∘ 𝐻)
2420, 23eqsstri 4015 . . . . . . . . 9 π‘Š βŠ† dom (𝐹 ∘ 𝐻)
2516, 3fcod 6740 . . . . . . . . 9 (πœ‘ β†’ (𝐹 ∘ 𝐻):(1...𝑀)⟢𝐡)
2624, 25fssdm 6734 . . . . . . . 8 (πœ‘ β†’ π‘Š βŠ† (1...𝑀))
27 ssfi 9169 . . . . . . . 8 (((1...𝑀) ∈ Fin ∧ π‘Š βŠ† (1...𝑀)) β†’ π‘Š ∈ Fin)
2822, 26, 27sylancr 587 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Fin)
2928ad2antrr 724 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š ∈ Fin)
301ad2antrr 724 . . . . . . . 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 724 . . . . . . . 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 6844 . . . . . . . 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 584 . . . . . . 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 6055 . . . . . . . . . 10 (𝐻 β€œ π‘Š) = (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 ))
3516, 15fexd 7225 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹 ∈ V)
36 ovex 7438 . . . . . . . . . . . . . 14 (1...𝑀) ∈ V
37 fex 7224 . . . . . . . . . . . . . 14 ((𝐻:(1...𝑀)⟢𝐴 ∧ (1...𝑀) ∈ V) β†’ 𝐻 ∈ V)
383, 36, 37sylancl 586 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐻 ∈ V)
3935, 38jca 512 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐹 ∈ V ∧ 𝐻 ∈ V))
40 f1fun 6786 . . . . . . . . . . . . . 14 (𝐻:(1...𝑀)–1-1→𝐴 β†’ Fun 𝐻)
411, 40syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ Fun 𝐻)
4241, 19jca 512 . . . . . . . . . . . 12 (πœ‘ β†’ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻))
43 imacosupp 8190 . . . . . . . . . . . 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 481 . . . . . . . . . 10 ((πœ‘ ∧ π‘Š β‰  βˆ…) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
4634, 45eqtrid 2784 . . . . . . . . 9 ((πœ‘ ∧ π‘Š β‰  βˆ…) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
4746adantr 481 . . . . . . . 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 6827 . . . . . . 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 14309 . . . . 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 6892 . . . 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 512 . . 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 6818 . . . 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 5856 . . . . . . 7 (𝑔 = (𝐻 ∘ 𝑓) β†’ (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑓)))
5554seqeq3d 13970 . . . . . 6 (𝑔 = (𝐻 ∘ 𝑓) β†’ seq1( + , (𝐹 ∘ 𝑔)) = seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓))))
5655fveq1d 6890 . . . . 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 2743 . . . 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 631 . . 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 3588 . 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 724 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐺 ∈ Mnd)
6115ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐴 ∈ 𝑉)
6216ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐹:𝐴⟢𝐡)
6317ad2antrr 724 . . . . . . 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 6841 . . . . . . . . . . . . 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 8958 . . . . . . . . . . . 12 ((𝐻 ∈ V ∧ 𝐻:(1...𝑀)–1-1-ontoβ†’ran 𝐻) β†’ (1...𝑀) β‰ˆ ran 𝐻)
675, 65, 66syl2anc 584 . . . . . . . . . . 11 (πœ‘ β†’ (1...𝑀) β‰ˆ ran 𝐻)
68 enfi 9186 . . . . . . . . . . 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 9263 . . . . . . . 8 (πœ‘ β†’ (𝐹 supp 0 ) ∈ Fin)
7271ad2antrr 724 . . . . . . 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 3005 . . . . . . . . . 10 (π‘Š β‰  βˆ… ↔ ((𝐹 ∘ 𝐻) supp 0 ) β‰  βˆ…)
74 supp0cosupp0 8189 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ ((𝐹 supp 0 ) = βˆ… β†’ ((𝐹 ∘ 𝐻) supp 0 ) = βˆ…))
7574necon3d 2961 . . . . . . . . . . 11 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ (((𝐹 ∘ 𝐻) supp 0 ) β‰  βˆ… β†’ (𝐹 supp 0 ) β‰  βˆ…))
7635, 38, 75syl2anc 584 . . . . . . . . . 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 407 . . . . . . . 8 ((πœ‘ ∧ π‘Š β‰  βˆ…) β†’ (𝐹 supp 0 ) β‰  βˆ…)
7978adantr 481 . . . . . . 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 724 . . . . . . . 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 6722 . . . . . . . . 9 (πœ‘ β†’ ran 𝐻 βŠ† 𝐴)
8281ad2antrr 724 . . . . . . . 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 3991 . . . . . . 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 19766 . . . . . 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 6517 . . . . . 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 2732 . . . . . . 7 (𝐹 supp 0 ) = (𝐹 supp 0 )
88 simprl 769 . . . . . . 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 19765 . . . . . 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 2734 . . . . 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 281 . . . 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 1930 . . 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 6901 . . . 4 (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))β€˜(β™―β€˜π‘Š)) ∈ V
94 eqeq1 2736 . . . . . . 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 629 . . . . . 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 1924 . . . . 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 2744 . . . . 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 345 . . . 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 3595 . . 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 396  βˆ€wal 1539   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆƒ!weu 2562   β‰  wne 2940  Vcvv 3474   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   ∘ ccom 5679  β„©cio 6490  Fun wfun 6534  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540   Isom wiso 6541  (class class class)co 7405   supp csupp 8142   β‰ˆ cen 8932  Fincfn 8935  1c1 11107   < clt 11244  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-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-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:  gsumval3  19769
  Copyright terms: Public domain W3C validator