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Theorem efgsrel 19643
Description: The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
efgval.r ∼ = ( ~FG β€˜πΌ)
efgval2.m 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
efgval2.t 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
efgred.d 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
efgred.s 𝑆 = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))
Assertion
Ref Expression
efgsrel (𝐹 ∈ dom 𝑆 β†’ (πΉβ€˜0) ∼ (π‘†β€˜πΉ))
Distinct variable groups:   𝑦,𝑧   𝑑,𝑛,𝑣,𝑀,𝑦,𝑧,π‘š,π‘₯   π‘š,𝑀   π‘₯,𝑛,𝑀,𝑑,𝑣,𝑀   π‘˜,π‘š,𝑑,π‘₯,𝑇   π‘˜,𝑛,𝑣,𝑀,𝑦,𝑧,π‘Š,π‘š,𝑑,π‘₯   ∼ ,π‘š,𝑑,π‘₯,𝑦,𝑧   π‘š,𝐼,𝑛,𝑑,𝑣,𝑀,π‘₯,𝑦,𝑧   𝐷,π‘š,𝑑
Allowed substitution hints:   𝐷(π‘₯,𝑦,𝑧,𝑀,𝑣,π‘˜,𝑛)   ∼ (𝑀,𝑣,π‘˜,𝑛)   𝑆(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝑇(𝑦,𝑧,𝑀,𝑣,𝑛)   𝐹(π‘₯,𝑦,𝑧,𝑀,𝑣,𝑑,π‘˜,π‘š,𝑛)   𝐼(π‘˜)   𝑀(𝑦,𝑧,π‘˜)
Proof of Theorem efgsrel
Dummy variables π‘Ž 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . 6 π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
2 efgval.r . . . . . 6 ∼ = ( ~FG β€˜πΌ)
3 efgval2.m . . . . . 6 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
4 efgval2.t . . . . . 6 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
5 efgred.d . . . . . 6 𝐷 = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))
6 efgred.s . . . . . 6 𝑆 = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))
71, 2, 3, 4, 5, 6efgsdm 19639 . . . . 5 (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘Ž ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘Ž) ∈ ran (π‘‡β€˜(πΉβ€˜(π‘Ž βˆ’ 1)))))
87simp1bi 1143 . . . 4 (𝐹 ∈ dom 𝑆 β†’ 𝐹 ∈ (Word π‘Š βˆ– {βˆ…}))
9 eldifsn 4789 . . . . 5 (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ↔ (𝐹 ∈ Word π‘Š ∧ 𝐹 β‰  βˆ…))
10 lennncl 14488 . . . . 5 ((𝐹 ∈ Word π‘Š ∧ 𝐹 β‰  βˆ…) β†’ (β™―β€˜πΉ) ∈ β„•)
119, 10sylbi 216 . . . 4 (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) β†’ (β™―β€˜πΉ) ∈ β„•)
12 fzo0end 13728 . . . 4 ((β™―β€˜πΉ) ∈ β„• β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
138, 11, 123syl 18 . . 3 (𝐹 ∈ dom 𝑆 β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
14 nnm1nn0 12517 . . . . 5 ((β™―β€˜πΉ) ∈ β„• β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0)
158, 11, 143syl 18 . . . 4 (𝐹 ∈ dom 𝑆 β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0)
16 eleq1 2819 . . . . . . 7 (π‘Ž = 0 β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) ↔ 0 ∈ (0..^(β™―β€˜πΉ))))
17 fveq2 6890 . . . . . . . 8 (π‘Ž = 0 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜0))
1817breq2d 5159 . . . . . . 7 (π‘Ž = 0 β†’ ((πΉβ€˜0) ∼ (πΉβ€˜π‘Ž) ↔ (πΉβ€˜0) ∼ (πΉβ€˜0)))
1916, 18imbi12d 343 . . . . . 6 (π‘Ž = 0 β†’ ((π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž)) ↔ (0 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜0))))
2019imbi2d 339 . . . . 5 (π‘Ž = 0 β†’ ((𝐹 ∈ dom 𝑆 β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž))) ↔ (𝐹 ∈ dom 𝑆 β†’ (0 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜0)))))
21 eleq1 2819 . . . . . . 7 (π‘Ž = 𝑖 β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) ↔ 𝑖 ∈ (0..^(β™―β€˜πΉ))))
22 fveq2 6890 . . . . . . . 8 (π‘Ž = 𝑖 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘–))
2322breq2d 5159 . . . . . . 7 (π‘Ž = 𝑖 β†’ ((πΉβ€˜0) ∼ (πΉβ€˜π‘Ž) ↔ (πΉβ€˜0) ∼ (πΉβ€˜π‘–)))
2421, 23imbi12d 343 . . . . . 6 (π‘Ž = 𝑖 β†’ ((π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž)) ↔ (𝑖 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–))))
2524imbi2d 339 . . . . 5 (π‘Ž = 𝑖 β†’ ((𝐹 ∈ dom 𝑆 β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž))) ↔ (𝐹 ∈ dom 𝑆 β†’ (𝑖 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–)))))
26 eleq1 2819 . . . . . . 7 (π‘Ž = (𝑖 + 1) β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) ↔ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))))
27 fveq2 6890 . . . . . . . 8 (π‘Ž = (𝑖 + 1) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜(𝑖 + 1)))
2827breq2d 5159 . . . . . . 7 (π‘Ž = (𝑖 + 1) β†’ ((πΉβ€˜0) ∼ (πΉβ€˜π‘Ž) ↔ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1))))
2926, 28imbi12d 343 . . . . . 6 (π‘Ž = (𝑖 + 1) β†’ ((π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž)) ↔ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1)))))
3029imbi2d 339 . . . . 5 (π‘Ž = (𝑖 + 1) β†’ ((𝐹 ∈ dom 𝑆 β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž))) ↔ (𝐹 ∈ dom 𝑆 β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1))))))
31 eleq1 2819 . . . . . . 7 (π‘Ž = ((β™―β€˜πΉ) βˆ’ 1) β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) ↔ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ))))
32 fveq2 6890 . . . . . . . 8 (π‘Ž = ((β™―β€˜πΉ) βˆ’ 1) β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)))
3332breq2d 5159 . . . . . . 7 (π‘Ž = ((β™―β€˜πΉ) βˆ’ 1) β†’ ((πΉβ€˜0) ∼ (πΉβ€˜π‘Ž) ↔ (πΉβ€˜0) ∼ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))))
3431, 33imbi12d 343 . . . . . 6 (π‘Ž = ((β™―β€˜πΉ) βˆ’ 1) β†’ ((π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž)) ↔ (((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)))))
3534imbi2d 339 . . . . 5 (π‘Ž = ((β™―β€˜πΉ) βˆ’ 1) β†’ ((𝐹 ∈ dom 𝑆 β†’ (π‘Ž ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘Ž))) ↔ (𝐹 ∈ dom 𝑆 β†’ (((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))))))
361, 2efger 19627 . . . . . . . 8 ∼ Er π‘Š
3736a1i 11 . . . . . . 7 ((𝐹 ∈ dom 𝑆 ∧ 0 ∈ (0..^(β™―β€˜πΉ))) β†’ ∼ Er π‘Š)
38 eldifi 4125 . . . . . . . . 9 (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) β†’ 𝐹 ∈ Word π‘Š)
39 wrdf 14473 . . . . . . . . 9 (𝐹 ∈ Word π‘Š β†’ 𝐹:(0..^(β™―β€˜πΉ))βŸΆπ‘Š)
408, 38, 393syl 18 . . . . . . . 8 (𝐹 ∈ dom 𝑆 β†’ 𝐹:(0..^(β™―β€˜πΉ))βŸΆπ‘Š)
4140ffvelcdmda 7085 . . . . . . 7 ((𝐹 ∈ dom 𝑆 ∧ 0 ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜0) ∈ π‘Š)
4237, 41erref 8725 . . . . . 6 ((𝐹 ∈ dom 𝑆 ∧ 0 ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜0) ∼ (πΉβ€˜0))
4342ex 411 . . . . 5 (𝐹 ∈ dom 𝑆 β†’ (0 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜0)))
44 elnn0uz 12871 . . . . . . . . . . . 12 (𝑖 ∈ β„•0 ↔ 𝑖 ∈ (β„€β‰₯β€˜0))
45 peano2fzor 13743 . . . . . . . . . . . 12 ((𝑖 ∈ (β„€β‰₯β€˜0) ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ 𝑖 ∈ (0..^(β™―β€˜πΉ)))
4644, 45sylanb 579 . . . . . . . . . . 11 ((𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ 𝑖 ∈ (0..^(β™―β€˜πΉ)))
47463adant1 1128 . . . . . . . . . 10 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ 𝑖 ∈ (0..^(β™―β€˜πΉ)))
48473expia 1119 . . . . . . . . 9 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0) β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ 𝑖 ∈ (0..^(β™―β€˜πΉ))))
4948imim1d 82 . . . . . . . 8 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0) β†’ ((𝑖 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–)) β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–))))
50403ad2ant1 1131 . . . . . . . . . . . . 13 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ 𝐹:(0..^(β™―β€˜πΉ))βŸΆπ‘Š)
5150, 47ffvelcdmd 7086 . . . . . . . . . . . 12 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜π‘–) ∈ π‘Š)
52 fvoveq1 7434 . . . . . . . . . . . . . . . . 17 (π‘Ž = (𝑖 + 1) β†’ (πΉβ€˜(π‘Ž βˆ’ 1)) = (πΉβ€˜((𝑖 + 1) βˆ’ 1)))
5352fveq2d 6894 . . . . . . . . . . . . . . . 16 (π‘Ž = (𝑖 + 1) β†’ (π‘‡β€˜(πΉβ€˜(π‘Ž βˆ’ 1))) = (π‘‡β€˜(πΉβ€˜((𝑖 + 1) βˆ’ 1))))
5453rneqd 5936 . . . . . . . . . . . . . . 15 (π‘Ž = (𝑖 + 1) β†’ ran (π‘‡β€˜(πΉβ€˜(π‘Ž βˆ’ 1))) = ran (π‘‡β€˜(πΉβ€˜((𝑖 + 1) βˆ’ 1))))
5527, 54eleq12d 2825 . . . . . . . . . . . . . 14 (π‘Ž = (𝑖 + 1) β†’ ((πΉβ€˜π‘Ž) ∈ ran (π‘‡β€˜(πΉβ€˜(π‘Ž βˆ’ 1))) ↔ (πΉβ€˜(𝑖 + 1)) ∈ ran (π‘‡β€˜(πΉβ€˜((𝑖 + 1) βˆ’ 1)))))
567simp3bi 1145 . . . . . . . . . . . . . . 15 (𝐹 ∈ dom 𝑆 β†’ βˆ€π‘Ž ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘Ž) ∈ ran (π‘‡β€˜(πΉβ€˜(π‘Ž βˆ’ 1))))
57563ad2ant1 1131 . . . . . . . . . . . . . 14 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ βˆ€π‘Ž ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘Ž) ∈ ran (π‘‡β€˜(πΉβ€˜(π‘Ž βˆ’ 1))))
58 nn0p1nn 12515 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ β„•0 β†’ (𝑖 + 1) ∈ β„•)
59583ad2ant2 1132 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (𝑖 + 1) ∈ β„•)
60 nnuz 12869 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
6159, 60eleqtrdi 2841 . . . . . . . . . . . . . . 15 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (𝑖 + 1) ∈ (β„€β‰₯β€˜1))
62 elfzolt2b 13647 . . . . . . . . . . . . . . . 16 ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (𝑖 + 1) ∈ ((𝑖 + 1)..^(β™―β€˜πΉ)))
63623ad2ant3 1133 . . . . . . . . . . . . . . 15 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (𝑖 + 1) ∈ ((𝑖 + 1)..^(β™―β€˜πΉ)))
64 elfzo3 13653 . . . . . . . . . . . . . . 15 ((𝑖 + 1) ∈ (1..^(β™―β€˜πΉ)) ↔ ((𝑖 + 1) ∈ (β„€β‰₯β€˜1) ∧ (𝑖 + 1) ∈ ((𝑖 + 1)..^(β™―β€˜πΉ))))
6561, 63, 64sylanbrc 581 . . . . . . . . . . . . . 14 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (𝑖 + 1) ∈ (1..^(β™―β€˜πΉ)))
6655, 57, 65rspcdva 3612 . . . . . . . . . . . . 13 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜(𝑖 + 1)) ∈ ran (π‘‡β€˜(πΉβ€˜((𝑖 + 1) βˆ’ 1))))
67 nn0cn 12486 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ β„•0 β†’ 𝑖 ∈ β„‚)
68673ad2ant2 1132 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ 𝑖 ∈ β„‚)
69 ax-1cn 11170 . . . . . . . . . . . . . . . . 17 1 ∈ β„‚
70 pncan 11470 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑖 + 1) βˆ’ 1) = 𝑖)
7168, 69, 70sylancl 584 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ ((𝑖 + 1) βˆ’ 1) = 𝑖)
7271fveq2d 6894 . . . . . . . . . . . . . . 15 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜((𝑖 + 1) βˆ’ 1)) = (πΉβ€˜π‘–))
7372fveq2d 6894 . . . . . . . . . . . . . 14 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (π‘‡β€˜(πΉβ€˜((𝑖 + 1) βˆ’ 1))) = (π‘‡β€˜(πΉβ€˜π‘–)))
7473rneqd 5936 . . . . . . . . . . . . 13 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ ran (π‘‡β€˜(πΉβ€˜((𝑖 + 1) βˆ’ 1))) = ran (π‘‡β€˜(πΉβ€˜π‘–)))
7566, 74eleqtrd 2833 . . . . . . . . . . . 12 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜(𝑖 + 1)) ∈ ran (π‘‡β€˜(πΉβ€˜π‘–)))
761, 2, 3, 4efgi2 19634 . . . . . . . . . . . 12 (((πΉβ€˜π‘–) ∈ π‘Š ∧ (πΉβ€˜(𝑖 + 1)) ∈ ran (π‘‡β€˜(πΉβ€˜π‘–))) β†’ (πΉβ€˜π‘–) ∼ (πΉβ€˜(𝑖 + 1)))
7751, 75, 76syl2anc 582 . . . . . . . . . . 11 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (πΉβ€˜π‘–) ∼ (πΉβ€˜(𝑖 + 1)))
7836a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ ∼ Er π‘Š)
7978ertr 8720 . . . . . . . . . . 11 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ (((πΉβ€˜0) ∼ (πΉβ€˜π‘–) ∧ (πΉβ€˜π‘–) ∼ (πΉβ€˜(𝑖 + 1))) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1))))
8077, 79mpan2d 690 . . . . . . . . . 10 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0 ∧ (𝑖 + 1) ∈ (0..^(β™―β€˜πΉ))) β†’ ((πΉβ€˜0) ∼ (πΉβ€˜π‘–) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1))))
81803expia 1119 . . . . . . . . 9 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0) β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ ((πΉβ€˜0) ∼ (πΉβ€˜π‘–) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1)))))
8281a2d 29 . . . . . . . 8 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0) β†’ (((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–)) β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1)))))
8349, 82syld 47 . . . . . . 7 ((𝐹 ∈ dom 𝑆 ∧ 𝑖 ∈ β„•0) β†’ ((𝑖 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–)) β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1)))))
8483expcom 412 . . . . . 6 (𝑖 ∈ β„•0 β†’ (𝐹 ∈ dom 𝑆 β†’ ((𝑖 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–)) β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1))))))
8584a2d 29 . . . . 5 (𝑖 ∈ β„•0 β†’ ((𝐹 ∈ dom 𝑆 β†’ (𝑖 ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜π‘–))) β†’ (𝐹 ∈ dom 𝑆 β†’ ((𝑖 + 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜(𝑖 + 1))))))
8620, 25, 30, 35, 43, 85nn0ind 12661 . . . 4 (((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0 β†’ (𝐹 ∈ dom 𝑆 β†’ (((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)))))
8715, 86mpcom 38 . . 3 (𝐹 ∈ dom 𝑆 β†’ (((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)) β†’ (πΉβ€˜0) ∼ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1))))
8813, 87mpd 15 . 2 (𝐹 ∈ dom 𝑆 β†’ (πΉβ€˜0) ∼ (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)))
891, 2, 3, 4, 5, 6efgsval 19640 . 2 (𝐹 ∈ dom 𝑆 β†’ (π‘†β€˜πΉ) = (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)))
9088, 89breqtrrd 5175 1 (𝐹 ∈ dom 𝑆 β†’ (πΉβ€˜0) ∼ (π‘†β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  {crab 3430   βˆ– cdif 3944  βˆ…c0 4321  {csn 4627  βŸ¨cop 4633  βŸ¨cotp 4635  βˆͺ ciun 4996   class class class wbr 5147   ↦ cmpt 5230   I cid 5572   Γ— cxp 5673  dom cdm 5675  ran crn 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1oc1o 8461  2oc2o 8462   Er wer 8702  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115   βˆ’ cmin 11448  β„•cn 12216  β„•0cn0 12476  β„€β‰₯cuz 12826  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468   splice csplice 14703  βŸ¨β€œcs2 14796   ~FG cefg 19615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-ec 8707  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625  df-splice 14704  df-s2 14803  df-efg 19618
This theorem is referenced by:  efgredeu  19661  efgred2  19662
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