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Theorem liminfequzmpt2 45102
Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
liminfequzmpt2.j β„²π‘—πœ‘
liminfequzmpt2.o Ⅎ𝑗𝐴
liminfequzmpt2.p Ⅎ𝑗𝐡
liminfequzmpt2.a 𝐴 = (β„€β‰₯β€˜π‘€)
liminfequzmpt2.b 𝐡 = (β„€β‰₯β€˜π‘)
liminfequzmpt2.k (πœ‘ β†’ 𝐾 ∈ 𝐴)
liminfequzmpt2.e (πœ‘ β†’ 𝐾 ∈ 𝐡)
liminfequzmpt2.c ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐢 ∈ 𝑉)
Assertion
Ref Expression
liminfequzmpt2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)))
Distinct variable group:   𝑗,𝐾
Allowed substitution hints:   πœ‘(𝑗)   𝐴(𝑗)   𝐡(𝑗)   𝐢(𝑗)   𝑀(𝑗)   𝑁(𝑗)   𝑉(𝑗)

Proof of Theorem liminfequzmpt2
StepHypRef Expression
1 liminfequzmpt2.j . . . . . . . . 9 β„²π‘—πœ‘
2 liminfequzmpt2.a . . . . . . . . . . . . . . 15 𝐴 = (β„€β‰₯β€˜π‘€)
3 liminfequzmpt2.k . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐾 ∈ 𝐴)
42, 3uzssd2 44722 . . . . . . . . . . . . . 14 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐴)
54adantr 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐴)
6 simpr 484 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ (β„€β‰₯β€˜πΎ))
75, 6sseldd 3979 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ 𝐴)
8 liminfequzmpt2.c . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐢 ∈ 𝑉)
98elexd 3490 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐢 ∈ V)
107, 9jca 511 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (𝑗 ∈ 𝐴 ∧ 𝐢 ∈ V))
11 rabid 3447 . . . . . . . . . . 11 (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↔ (𝑗 ∈ 𝐴 ∧ 𝐢 ∈ V))
1210, 11sylibr 233 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
1312ex 412 . . . . . . . . 9 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) β†’ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}))
141, 13ralrimi 3249 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
15 nfcv 2898 . . . . . . . . 9 Ⅎ𝑗(β„€β‰₯β€˜πΎ)
16 nfrab1 3446 . . . . . . . . 9 Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}
1715, 16dfss3f 3969 . . . . . . . 8 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↔ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
1814, 17sylibr 233 . . . . . . 7 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
1916, 15resmptf 6037 . . . . . . 7 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} β†’ ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
2018, 19syl 17 . . . . . 6 (πœ‘ β†’ ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
2120eqcomd 2733 . . . . 5 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)))
2221fveq2d 6895 . . . 4 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)) = (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))))
232, 3eluzelz2d 44718 . . . . 5 (πœ‘ β†’ 𝐾 ∈ β„€)
24 eqid 2727 . . . . 5 (β„€β‰₯β€˜πΎ) = (β„€β‰₯β€˜πΎ)
25 liminfequzmpt2.o . . . . . . . 8 Ⅎ𝑗𝐴
262fvexi 6905 . . . . . . . 8 𝐴 ∈ V
2725, 26rabexf 44423 . . . . . . 7 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ∈ V
2816, 27mptexf 44535 . . . . . 6 (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V
2928a1i 11 . . . . 5 (πœ‘ β†’ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V)
30 eqid 2727 . . . . . . . 8 (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)
3116, 30dmmptssf 44529 . . . . . . 7 dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}
3225ssrab2f 44406 . . . . . . . 8 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} βŠ† 𝐴
33 uzssz 12865 . . . . . . . . 9 (β„€β‰₯β€˜π‘€) βŠ† β„€
342, 33eqsstri 4012 . . . . . . . 8 𝐴 βŠ† β„€
3532, 34sstri 3987 . . . . . . 7 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} βŠ† β„€
3631, 35sstri 3987 . . . . . 6 dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€
3736a1i 11 . . . . 5 (πœ‘ β†’ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€)
3823, 24, 29, 37liminfresuz2 45098 . . . 4 (πœ‘ β†’ (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)))
3922, 38eqtr2d 2768 . . 3 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)))
40 liminfequzmpt2.b . . . . . . . . . . . . . . 15 𝐡 = (β„€β‰₯β€˜π‘)
41 liminfequzmpt2.e . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐾 ∈ 𝐡)
4240, 41uzssd2 44722 . . . . . . . . . . . . . 14 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐡)
4342adantr 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐡)
4443, 6sseldd 3979 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ 𝐡)
4544, 9jca 511 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (𝑗 ∈ 𝐡 ∧ 𝐢 ∈ V))
46 rabid 3447 . . . . . . . . . . 11 (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↔ (𝑗 ∈ 𝐡 ∧ 𝐢 ∈ V))
4745, 46sylibr 233 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
4847ex 412 . . . . . . . . 9 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) β†’ 𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}))
491, 48ralrimi 3249 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
50 nfrab1 3446 . . . . . . . . 9 Ⅎ𝑗{𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}
5115, 50dfss3f 3969 . . . . . . . 8 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↔ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
5249, 51sylibr 233 . . . . . . 7 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
5350, 15resmptf 6037 . . . . . . 7 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} β†’ ((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
5452, 53syl 17 . . . . . 6 (πœ‘ β†’ ((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
5554eqcomd 2733 . . . . 5 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢) = ((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)))
5655fveq2d 6895 . . . 4 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)) = (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))))
57 liminfequzmpt2.p . . . . . . . 8 Ⅎ𝑗𝐡
5840fvexi 6905 . . . . . . . 8 𝐡 ∈ V
5957, 58rabexf 44423 . . . . . . 7 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ∈ V
6050, 59mptexf 44535 . . . . . 6 (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V
6160a1i 11 . . . . 5 (πœ‘ β†’ (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V)
62 eqid 2727 . . . . . . . 8 (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)
6350, 62dmmptssf 44529 . . . . . . 7 dom (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}
6457ssrab2f 44406 . . . . . . . 8 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} βŠ† 𝐡
65 uzssz 12865 . . . . . . . . 9 (β„€β‰₯β€˜π‘) βŠ† β„€
6640, 65eqsstri 4012 . . . . . . . 8 𝐡 βŠ† β„€
6764, 66sstri 3987 . . . . . . 7 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} βŠ† β„€
6863, 67sstri 3987 . . . . . 6 dom (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€
6968a1i 11 . . . . 5 (πœ‘ β†’ dom (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€)
7023, 24, 61, 69liminfresuz2 45098 . . . 4 (πœ‘ β†’ (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)))
7156, 70eqtr2d 2768 . . 3 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)))
7239, 71eqtr4d 2770 . 2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)))
73 eqid 2727 . . . . 5 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}
7425, 73mptssid 44539 . . . 4 (𝑗 ∈ 𝐴 ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)
7574fveq2i 6894 . . 3 (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢))
7675a1i 11 . 2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)))
77 eqid 2727 . . . . 5 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} = {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}
7857, 77mptssid 44539 . . . 4 (𝑗 ∈ 𝐡 ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)
7978fveq2i 6894 . . 3 (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢))
8079a1i 11 . 2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)))
8172, 76, 803eqtr4d 2777 1 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534  β„²wnf 1778   ∈ wcel 2099  β„²wnfc 2878  βˆ€wral 3056  {crab 3427  Vcvv 3469   βŠ† wss 3944   ↦ cmpt 5225  dom cdm 5672   β†Ύ cres 5674  β€˜cfv 6542  β„€cz 12580  β„€β‰₯cuz 12844  lim infclsi 45062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9457  df-inf 9458  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-ico 13354  df-liminf 45063
This theorem is referenced by:  smfliminfmpt  46143
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