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Theorem nninffeq 14854
Description: Equality of two functions on β„•βˆž which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, (πœ‘ β†’ βˆ€π‘› ∈ suc Ο‰...). (Contributed by Jim Kingdon, 4-Aug-2023.)
Hypotheses
Ref Expression
nninffeq.f (πœ‘ β†’ 𝐹:β„•βˆžβŸΆβ„•0)
nninffeq.g (πœ‘ β†’ 𝐺:β„•βˆžβŸΆβ„•0)
nninffeq.oo (πœ‘ β†’ (πΉβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)))
nninffeq.n (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))))
Assertion
Ref Expression
nninffeq (πœ‘ β†’ 𝐹 = 𝐺)
Distinct variable groups:   𝑖,𝐹,𝑛,π‘₯   𝑖,𝐺,𝑛,π‘₯   πœ‘,𝑖,𝑛,π‘₯

Proof of Theorem nninffeq
Dummy variables 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nninffeq.f . . 3 (πœ‘ β†’ 𝐹:β„•βˆžβŸΆβ„•0)
21ffnd 5368 . 2 (πœ‘ β†’ 𝐹 Fn β„•βˆž)
3 nninffeq.g . . 3 (πœ‘ β†’ 𝐺:β„•βˆžβŸΆβ„•0)
43ffnd 5368 . 2 (πœ‘ β†’ 𝐺 Fn β„•βˆž)
5 eqid 2177 . . . . . . . 8 (π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…)) = (π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))
6 fveq2 5517 . . . . . . . . . 10 (π‘₯ = 𝑧 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘§))
7 fveq2 5517 . . . . . . . . . 10 (π‘₯ = 𝑧 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘§))
86, 7eqeq12d 2192 . . . . . . . . 9 (π‘₯ = 𝑧 β†’ ((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) ↔ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)))
98ifbid 3557 . . . . . . . 8 (π‘₯ = 𝑧 β†’ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…) = if((πΉβ€˜π‘§) = (πΊβ€˜π‘§), 1o, βˆ…))
10 simpr 110 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ 𝑧 ∈ β„•βˆž)
11 1onn 6523 . . . . . . . . . 10 1o ∈ Ο‰
1211a1i 9 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ 1o ∈ Ο‰)
13 peano1 4595 . . . . . . . . . 10 βˆ… ∈ Ο‰
1413a1i 9 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ βˆ… ∈ Ο‰)
151ffvelcdmda 5653 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ (πΉβ€˜π‘§) ∈ β„•0)
1615nn0zd 9375 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ (πΉβ€˜π‘§) ∈ β„€)
173ffvelcdmda 5653 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ (πΊβ€˜π‘§) ∈ β„•0)
1817nn0zd 9375 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ (πΊβ€˜π‘§) ∈ β„€)
19 zdceq 9330 . . . . . . . . . 10 (((πΉβ€˜π‘§) ∈ β„€ ∧ (πΊβ€˜π‘§) ∈ β„€) β†’ DECID (πΉβ€˜π‘§) = (πΊβ€˜π‘§))
2016, 18, 19syl2anc 411 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ DECID (πΉβ€˜π‘§) = (πΊβ€˜π‘§))
2112, 14, 20ifcldcd 3572 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ if((πΉβ€˜π‘§) = (πΊβ€˜π‘§), 1o, βˆ…) ∈ Ο‰)
225, 9, 10, 21fvmptd3 5611 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜π‘§) = if((πΉβ€˜π‘§) = (πΊβ€˜π‘§), 1o, βˆ…))
23 1lt2o 6445 . . . . . . . . . . . . 13 1o ∈ 2o
2423a1i 9 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ 1o ∈ 2o)
25 0lt2o 6444 . . . . . . . . . . . . 13 βˆ… ∈ 2o
2625a1i 9 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ βˆ… ∈ 2o)
271ffvelcdmda 5653 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ (πΉβ€˜π‘₯) ∈ β„•0)
2827nn0zd 9375 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ (πΉβ€˜π‘₯) ∈ β„€)
293ffvelcdmda 5653 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ (πΊβ€˜π‘₯) ∈ β„•0)
3029nn0zd 9375 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ (πΊβ€˜π‘₯) ∈ β„€)
31 zdceq 9330 . . . . . . . . . . . . 13 (((πΉβ€˜π‘₯) ∈ β„€ ∧ (πΊβ€˜π‘₯) ∈ β„€) β†’ DECID (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))
3228, 30, 31syl2anc 411 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ DECID (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))
3324, 26, 32ifcldcd 3572 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ β„•βˆž) β†’ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…) ∈ 2o)
3433fmpttd 5673 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…)):β„•βˆžβŸΆ2o)
35 2onn 6524 . . . . . . . . . . . 12 2o ∈ Ο‰
3635elexi 2751 . . . . . . . . . . 11 2o ∈ V
37 nninfex 7122 . . . . . . . . . . 11 β„•βˆž ∈ V
3836, 37elmap 6679 . . . . . . . . . 10 ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…)) ∈ (2o β†‘π‘š β„•βˆž) ↔ (π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…)):β„•βˆžβŸΆ2o)
3934, 38sylibr 134 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…)) ∈ (2o β†‘π‘š β„•βˆž))
40 fveq2 5517 . . . . . . . . . . . . 13 (π‘₯ = (𝑀 ∈ Ο‰ ↦ 1o) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)))
41 fveq2 5517 . . . . . . . . . . . . 13 (π‘₯ = (𝑀 ∈ Ο‰ ↦ 1o) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o)))
4240, 41eqeq12d 2192 . . . . . . . . . . . 12 (π‘₯ = (𝑀 ∈ Ο‰ ↦ 1o) β†’ ((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) ↔ (πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o))))
4342ifbid 3557 . . . . . . . . . . 11 (π‘₯ = (𝑀 ∈ Ο‰ ↦ 1o) β†’ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…) = if((πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o)), 1o, βˆ…))
44 infnninf 7124 . . . . . . . . . . . 12 (𝑀 ∈ Ο‰ ↦ 1o) ∈ β„•βˆž
4544a1i 9 . . . . . . . . . . 11 (πœ‘ β†’ (𝑀 ∈ Ο‰ ↦ 1o) ∈ β„•βˆž)
46 nninffeq.oo . . . . . . . . . . . . . 14 (πœ‘ β†’ (πΉβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)))
47 eqidd 2178 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑀 β†’ 1o = 1o)
4847cbvmptv 4101 . . . . . . . . . . . . . . 15 (π‘₯ ∈ Ο‰ ↦ 1o) = (𝑀 ∈ Ο‰ ↦ 1o)
4948fveq2i 5520 . . . . . . . . . . . . . 14 (πΉβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)) = (πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o))
5048fveq2i 5520 . . . . . . . . . . . . . 14 (πΊβ€˜(π‘₯ ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o))
5146, 49, 503eqtr3g 2233 . . . . . . . . . . . . 13 (πœ‘ β†’ (πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o)))
5251iftrued 3543 . . . . . . . . . . . 12 (πœ‘ β†’ if((πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o)), 1o, βˆ…) = 1o)
5352, 11eqeltrdi 2268 . . . . . . . . . . 11 (πœ‘ β†’ if((πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o)), 1o, βˆ…) ∈ Ο‰)
545, 43, 45, 53fvmptd3 5611 . . . . . . . . . 10 (πœ‘ β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑀 ∈ Ο‰ ↦ 1o)) = if((πΉβ€˜(𝑀 ∈ Ο‰ ↦ 1o)) = (πΊβ€˜(𝑀 ∈ Ο‰ ↦ 1o)), 1o, βˆ…))
5554, 52eqtrd 2210 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑀 ∈ Ο‰ ↦ 1o)) = 1o)
56 nninffeq.n . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))))
57 fveq2 5517 . . . . . . . . . . . . . . . 16 (π‘₯ = (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))))
58 fveq2 5517 . . . . . . . . . . . . . . . 16 (π‘₯ = (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))))
5957, 58eqeq12d 2192 . . . . . . . . . . . . . . 15 (π‘₯ = (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)) β†’ ((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) ↔ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))))
6059ifbid 3557 . . . . . . . . . . . . . 14 (π‘₯ = (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)) β†’ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…) = if((πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))), 1o, βˆ…))
61 nnnninf 7126 . . . . . . . . . . . . . . 15 (𝑛 ∈ Ο‰ β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)) ∈ β„•βˆž)
6261ad2antlr 489 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))) β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)) ∈ β„•βˆž)
63 simpr 110 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))) β†’ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))))
6463iftrued 3543 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))) β†’ if((πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))), 1o, βˆ…) = 1o)
6564, 11eqeltrdi 2268 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))) β†’ if((πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))), 1o, βˆ…) ∈ Ο‰)
665, 60, 62, 65fvmptd3 5611 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))) β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = if((πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))), 1o, βˆ…))
6766, 64eqtrd 2210 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…)))) β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = 1o)
6867ex 115 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = 1o))
6968ralimdva 2544 . . . . . . . . . 10 (πœ‘ β†’ (βˆ€π‘› ∈ Ο‰ (πΉβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = (πΊβ€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) β†’ βˆ€π‘› ∈ Ο‰ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = 1o))
7056, 69mpd 13 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜(𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑛, 1o, βˆ…))) = 1o)
7139, 55, 70nninfall 14843 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘§ ∈ β„•βˆž ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜π‘§) = 1o)
7271r19.21bi 2565 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ ((π‘₯ ∈ β„•βˆž ↦ if((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯), 1o, βˆ…))β€˜π‘§) = 1o)
7322, 72eqtr3d 2212 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ if((πΉβ€˜π‘§) = (πΊβ€˜π‘§), 1o, βˆ…) = 1o)
7473adantr 276 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ β„•βˆž) ∧ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)) β†’ if((πΉβ€˜π‘§) = (πΊβ€˜π‘§), 1o, βˆ…) = 1o)
75 simpr 110 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ β„•βˆž) ∧ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)) β†’ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§))
7675iffalsed 3546 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ β„•βˆž) ∧ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)) β†’ if((πΉβ€˜π‘§) = (πΊβ€˜π‘§), 1o, βˆ…) = βˆ…)
7774, 76eqtr3d 2212 . . . 4 (((πœ‘ ∧ 𝑧 ∈ β„•βˆž) ∧ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)) β†’ 1o = βˆ…)
78 1n0 6435 . . . . . 6 1o β‰  βˆ…
7978neii 2349 . . . . 5 Β¬ 1o = βˆ…
8079a1i 9 . . . 4 (((πœ‘ ∧ 𝑧 ∈ β„•βˆž) ∧ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)) β†’ Β¬ 1o = βˆ…)
8177, 80pm2.65da 661 . . 3 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ Β¬ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§))
82 exmiddc 836 . . . 4 (DECID (πΉβ€˜π‘§) = (πΊβ€˜π‘§) β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ∨ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)))
8320, 82syl 14 . . 3 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ∨ Β¬ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)))
8481, 83ecased 1349 . 2 ((πœ‘ ∧ 𝑧 ∈ β„•βˆž) β†’ (πΉβ€˜π‘§) = (πΊβ€˜π‘§))
852, 4, 84eqfnfvd 5618 1 (πœ‘ β†’ 𝐹 = 𝐺)
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆ…c0 3424  ifcif 3536   ↦ cmpt 4066  Ο‰com 4591  βŸΆwf 5214  β€˜cfv 5218  (class class class)co 5877  1oc1o 6412  2oc2o 6413   β†‘π‘š cmap 6650  β„•βˆžxnninf 7120  β„•0cn0 9178  β„€cz 9255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1o 6419  df-2o 6420  df-map 6652  df-nninf 7121  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256
This theorem is referenced by: (None)
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