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Theorem gsumval3eu 19686
Description: The group sum as defined in gsumval3a 19685 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
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))
gsumval3a.t (πœ‘ β†’ π‘Š ∈ Fin)
gsumval3a.n (πœ‘ β†’ π‘Š β‰  βˆ…)
gsumval3a.s (πœ‘ β†’ π‘Š βŠ† 𝐴)
Assertion
Ref Expression
gsumval3eu (πœ‘ β†’ βˆƒ!π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))
Distinct variable groups:   π‘₯,𝑓, +   𝐴,𝑓,π‘₯   πœ‘,𝑓,π‘₯   π‘₯, 0   𝑓,𝐺,π‘₯   π‘₯,𝑉   𝐡,𝑓,π‘₯   𝑓,𝐹,π‘₯   𝑓,π‘Š,π‘₯
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(π‘₯,𝑓)

Proof of Theorem gsumval3eu
Dummy variables 𝑔 π‘˜ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3a.n . . . . . 6 (πœ‘ β†’ π‘Š β‰  βˆ…)
21neneqd 2945 . . . . 5 (πœ‘ β†’ Β¬ π‘Š = βˆ…)
3 gsumval3a.t . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Fin)
4 fz1f1o 15600 . . . . . . 7 (π‘Š ∈ Fin β†’ (π‘Š = βˆ… ∨ ((β™―β€˜π‘Š) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)))
53, 4syl 17 . . . . . 6 (πœ‘ β†’ (π‘Š = βˆ… ∨ ((β™―β€˜π‘Š) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)))
65ord 863 . . . . 5 (πœ‘ β†’ (Β¬ π‘Š = βˆ… β†’ ((β™―β€˜π‘Š) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)))
72, 6mpd 15 . . . 4 (πœ‘ β†’ ((β™―β€˜π‘Š) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š))
87simprd 497 . . 3 (πœ‘ β†’ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
9 excom 2163 . . . 4 (βˆƒπ‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘“βˆƒπ‘₯(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))
10 exancom 1865 . . . . . 6 (βˆƒπ‘₯(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘₯(π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∧ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š))
11 fvex 6856 . . . . . . 7 (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∈ V
12 biidd 262 . . . . . . 7 (π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) β†’ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ↔ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š))
1311, 12ceqsexv 3493 . . . . . 6 (βˆƒπ‘₯(π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∧ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š) ↔ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
1410, 13bitri 275 . . . . 5 (βˆƒπ‘₯(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
1514exbii 1851 . . . 4 (βˆƒπ‘“βˆƒπ‘₯(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
169, 15bitri 275 . . 3 (βˆƒπ‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
178, 16sylibr 233 . 2 (πœ‘ β†’ βˆƒπ‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))
18 exdistrv 1960 . . . 4 (βˆƒπ‘“βˆƒπ‘”((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) ↔ (βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ βˆƒπ‘”(𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))))
19 an4 655 . . . . . 6 (((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š) ∧ (π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) ↔ ((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))))
20 gsumval3.g . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ Mnd)
2120adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝐺 ∈ Mnd)
22 gsumval3.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΊ)
23 gsumval3.p . . . . . . . . . . . 12 + = (+gβ€˜πΊ)
2422, 23mndcl 18569 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ + 𝑦) ∈ 𝐡)
25243expb 1121 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯ + 𝑦) ∈ 𝐡)
2621, 25sylan 581 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯ + 𝑦) ∈ 𝐡)
27 gsumval3.c . . . . . . . . . . . . 13 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
2827adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
2928sselda 3945 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘₯ ∈ ran 𝐹) β†’ π‘₯ ∈ (π‘β€˜ran 𝐹))
3029adantrr 716 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) β†’ π‘₯ ∈ (π‘β€˜ran 𝐹))
31 simprr 772 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) β†’ 𝑦 ∈ ran 𝐹)
32 gsumval3.z . . . . . . . . . . 11 𝑍 = (Cntzβ€˜πΊ)
3323, 32cntzi 19114 . . . . . . . . . 10 ((π‘₯ ∈ (π‘β€˜ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))
3430, 31, 33syl2anc 585 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))
3522, 23mndass 18570 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))
3621, 35sylan 581 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))
377simpld 496 . . . . . . . . . . 11 (πœ‘ β†’ (β™―β€˜π‘Š) ∈ β„•)
3837adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
39 nnuz 12811 . . . . . . . . . 10 β„• = (β„€β‰₯β€˜1)
4038, 39eleqtrdi 2844 . . . . . . . . 9 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜1))
41 gsumval3.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
4241adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝐹:𝐴⟢𝐡)
4342frnd 6677 . . . . . . . . 9 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ ran 𝐹 βŠ† 𝐡)
44 simprr 772 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
45 f1ocnv 6797 . . . . . . . . . . 11 (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š β†’ ◑𝑔:π‘Šβ€“1-1-ontoβ†’(1...(β™―β€˜π‘Š)))
4644, 45syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ ◑𝑔:π‘Šβ€“1-1-ontoβ†’(1...(β™―β€˜π‘Š)))
47 simprl 770 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
48 f1oco 6808 . . . . . . . . . 10 ((◑𝑔:π‘Šβ€“1-1-ontoβ†’(1...(β™―β€˜π‘Š)) ∧ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š) β†’ (◑𝑔 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(1...(β™―β€˜π‘Š)))
4946, 47, 48syl2anc 585 . . . . . . . . 9 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ (◑𝑔 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(1...(β™―β€˜π‘Š)))
50 f1of 6785 . . . . . . . . . . . 12 (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š β†’ 𝑔:(1...(β™―β€˜π‘Š))βŸΆπ‘Š)
5144, 50syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝑔:(1...(β™―β€˜π‘Š))βŸΆπ‘Š)
52 fvco3 6941 . . . . . . . . . . 11 ((𝑔:(1...(β™―β€˜π‘Š))βŸΆπ‘Š ∧ π‘₯ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑔)β€˜π‘₯) = (πΉβ€˜(π‘”β€˜π‘₯)))
5351, 52sylan 581 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘₯ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑔)β€˜π‘₯) = (πΉβ€˜(π‘”β€˜π‘₯)))
5442ffnd 6670 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝐹 Fn 𝐴)
55 gsumval3a.s . . . . . . . . . . . . . 14 (πœ‘ β†’ π‘Š βŠ† 𝐴)
5655adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ π‘Š βŠ† 𝐴)
5751, 56fssd 6687 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝑔:(1...(β™―β€˜π‘Š))⟢𝐴)
5857ffvelcdmda 7036 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘₯ ∈ (1...(β™―β€˜π‘Š))) β†’ (π‘”β€˜π‘₯) ∈ 𝐴)
59 fnfvelrn 7032 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (π‘”β€˜π‘₯) ∈ 𝐴) β†’ (πΉβ€˜(π‘”β€˜π‘₯)) ∈ ran 𝐹)
6054, 58, 59syl2an2r 684 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘₯ ∈ (1...(β™―β€˜π‘Š))) β†’ (πΉβ€˜(π‘”β€˜π‘₯)) ∈ ran 𝐹)
6153, 60eqeltrd 2834 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘₯ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑔)β€˜π‘₯) ∈ ran 𝐹)
62 f1of 6785 . . . . . . . . . . . . . . 15 (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š β†’ 𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š)
6347, 62syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ 𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š)
64 fvco3 6941 . . . . . . . . . . . . . 14 ((𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((◑𝑔 ∘ 𝑓)β€˜π‘˜) = (β—‘π‘”β€˜(π‘“β€˜π‘˜)))
6563, 64sylan 581 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((◑𝑔 ∘ 𝑓)β€˜π‘˜) = (β—‘π‘”β€˜(π‘“β€˜π‘˜)))
6665fveq2d 6847 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ (π‘”β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜)) = (π‘”β€˜(β—‘π‘”β€˜(π‘“β€˜π‘˜))))
6763ffvelcdmda 7036 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ (π‘“β€˜π‘˜) ∈ π‘Š)
68 f1ocnvfv2 7224 . . . . . . . . . . . . 13 ((𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ (π‘“β€˜π‘˜) ∈ π‘Š) β†’ (π‘”β€˜(β—‘π‘”β€˜(π‘“β€˜π‘˜))) = (π‘“β€˜π‘˜))
6944, 67, 68syl2an2r 684 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ (π‘”β€˜(β—‘π‘”β€˜(π‘“β€˜π‘˜))) = (π‘“β€˜π‘˜))
7066, 69eqtr2d 2774 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ (π‘“β€˜π‘˜) = (π‘”β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜)))
7170fveq2d 6847 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ (πΉβ€˜(π‘“β€˜π‘˜)) = (πΉβ€˜(π‘”β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜))))
72 fvco3 6941 . . . . . . . . . . 11 ((𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑓)β€˜π‘˜) = (πΉβ€˜(π‘“β€˜π‘˜)))
7363, 72sylan 581 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑓)β€˜π‘˜) = (πΉβ€˜(π‘“β€˜π‘˜)))
74 f1of 6785 . . . . . . . . . . . . 13 ((◑𝑔 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(1...(β™―β€˜π‘Š)) β†’ (◑𝑔 ∘ 𝑓):(1...(β™―β€˜π‘Š))⟢(1...(β™―β€˜π‘Š)))
7549, 74syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ (◑𝑔 ∘ 𝑓):(1...(β™―β€˜π‘Š))⟢(1...(β™―β€˜π‘Š)))
7675ffvelcdmda 7036 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((◑𝑔 ∘ 𝑓)β€˜π‘˜) ∈ (1...(β™―β€˜π‘Š)))
77 fvco3 6941 . . . . . . . . . . 11 ((𝑔:(1...(β™―β€˜π‘Š))⟢𝐴 ∧ ((◑𝑔 ∘ 𝑓)β€˜π‘˜) ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑔)β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜)) = (πΉβ€˜(π‘”β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜))))
7857, 76, 77syl2an2r 684 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑔)β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜)) = (πΉβ€˜(π‘”β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜))))
7971, 73, 783eqtr4d 2783 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) ∧ π‘˜ ∈ (1...(β™―β€˜π‘Š))) β†’ ((𝐹 ∘ 𝑓)β€˜π‘˜) = ((𝐹 ∘ 𝑔)β€˜((◑𝑔 ∘ 𝑓)β€˜π‘˜)))
8026, 34, 36, 40, 43, 49, 61, 79seqf1o 13955 . . . . . . . 8 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))
81 eqeq12 2750 . . . . . . . 8 ((π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š))) β†’ (π‘₯ = 𝑦 ↔ (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š))))
8280, 81syl5ibrcom 247 . . . . . . 7 ((πœ‘ ∧ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)) β†’ ((π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š))) β†’ π‘₯ = 𝑦))
8382expimpd 455 . . . . . 6 (πœ‘ β†’ (((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š) ∧ (π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) β†’ π‘₯ = 𝑦))
8419, 83biimtrrid 242 . . . . 5 (πœ‘ β†’ (((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) β†’ π‘₯ = 𝑦))
8584exlimdvv 1938 . . . 4 (πœ‘ β†’ (βˆƒπ‘“βˆƒπ‘”((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) β†’ π‘₯ = 𝑦))
8618, 85biimtrrid 242 . . 3 (πœ‘ β†’ ((βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ βˆƒπ‘”(𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) β†’ π‘₯ = 𝑦))
8786alrimivv 1932 . 2 (πœ‘ β†’ βˆ€π‘₯βˆ€π‘¦((βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ βˆƒπ‘”(𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) β†’ π‘₯ = 𝑦))
88 eqeq1 2737 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))
8988anbi2d 630 . . . . 5 (π‘₯ = 𝑦 β†’ ((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))
9089exbidv 1925 . . . 4 (π‘₯ = 𝑦 β†’ (βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)))))
91 f1oeq1 6773 . . . . . 6 (𝑓 = 𝑔 β†’ (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ↔ 𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š))
92 coeq2 5815 . . . . . . . . 9 (𝑓 = 𝑔 β†’ (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
9392seqeq3d 13920 . . . . . . . 8 (𝑓 = 𝑔 β†’ seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔)))
9493fveq1d 6845 . . . . . . 7 (𝑓 = 𝑔 β†’ (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))
9594eqeq2d 2744 . . . . . 6 (𝑓 = 𝑔 β†’ (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š))))
9691, 95anbi12d 632 . . . . 5 (𝑓 = 𝑔 β†’ ((𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ (𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))))
9796cbvexvw 2041 . . . 4 (βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘”(𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š))))
9890, 97bitrdi 287 . . 3 (π‘₯ = 𝑦 β†’ (βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ βˆƒπ‘”(𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))))
9998eu4 2612 . 2 (βˆƒ!π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ↔ (βˆƒπ‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ βˆ€π‘₯βˆ€π‘¦((βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))) ∧ βˆƒπ‘”(𝑔:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))β€˜(β™―β€˜π‘Š)))) β†’ π‘₯ = 𝑦)))
10017, 87, 99sylanbrc 584 1 (πœ‘ β†’ βˆƒ!π‘₯βˆƒπ‘“(𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š ∧ π‘₯ = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜π‘Š))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒ!weu 2563   β‰  wne 2940   βŠ† wss 3911  βˆ…c0 4283  β—‘ccnv 5633  ran crn 5635   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  1c1 11057  β„•cn 12158  β„€β‰₯cuz 12768  ...cfz 13430  seqcseq 13912  β™―chash 14236  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Mndcmnd 18561  Cntzccntz 19100
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-seq 13913  df-hash 14237  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-cntz 19102
This theorem is referenced by:  gsumval3lem2  19688
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