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Theorem summolem2 14294
Description: Lemma for summo 14295. (Contributed by Mario Carneiro, 3-Apr-2014.)
Hypotheses
Ref Expression
summo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
summo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
summo.3 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
Assertion
Ref Expression
summolem2 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Distinct variable groups:   𝑓,𝑘,𝑚,𝑛,𝑥,𝑦,𝐴   𝑓,𝐹,𝑘,𝑚,𝑛,𝑥,𝑦   𝑘,𝐺,𝑚,𝑛,𝑥,𝑦   𝜑,𝑘,𝑚,𝑛,𝑦   𝐵,𝑓,𝑚,𝑛,𝑥,𝑦   𝜑,𝑥,𝑓
Allowed substitution hints:   𝐵(𝑘)   𝐺(𝑓)

Proof of Theorem summolem2
Dummy variables 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . 5 (𝑚 = 𝑗 → (ℤ𝑚) = (ℤ𝑗))
21sseq2d 3596 . . . 4 (𝑚 = 𝑗 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑗)))
3 seqeq1 12666 . . . . 5 (𝑚 = 𝑗 → seq𝑚( + , 𝐹) = seq𝑗( + , 𝐹))
43breq1d 4593 . . . 4 (𝑚 = 𝑗 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑗( + , 𝐹) ⇝ 𝑥))
52, 4anbi12d 743 . . 3 (𝑚 = 𝑗 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)))
65cbvrexv 3148 . 2 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑗 ∈ ℤ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥))
7 simplrr 797 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑗( + , 𝐹) ⇝ 𝑥)
8 simplrl 796 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ (ℤ𝑗))
9 uzssz 11583 . . . . . . . . . . . . . 14 (ℤ𝑗) ⊆ ℤ
10 zssre 11261 . . . . . . . . . . . . . 14 ℤ ⊆ ℝ
119, 10sstri 3577 . . . . . . . . . . . . 13 (ℤ𝑗) ⊆ ℝ
128, 11syl6ss 3580 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ ℝ)
13 ltso 9997 . . . . . . . . . . . 12 < Or ℝ
14 soss 4977 . . . . . . . . . . . 12 (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
1512, 13, 14mpisyl 21 . . . . . . . . . . 11 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → < Or 𝐴)
16 fzfi 12633 . . . . . . . . . . . 12 (1...𝑚) ∈ Fin
17 ovex 6577 . . . . . . . . . . . . . . 15 (1...𝑚) ∈ V
1817f1oen 7862 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
1918ad2antll 761 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ≈ 𝐴)
2019ensymd 7893 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ≈ (1...𝑚))
21 enfii 8062 . . . . . . . . . . . 12 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
2216, 20, 21sylancr 694 . . . . . . . . . . 11 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ∈ Fin)
23 fz1iso 13103 . . . . . . . . . . 11 (( < Or 𝐴𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))
2415, 22, 23syl2anc 691 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))
25 summo.1 . . . . . . . . . . . . 13 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
26 simplll 794 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝜑)
27 summo.2 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2826, 27sylan 487 . . . . . . . . . . . . 13 (((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
29 summo.3 . . . . . . . . . . . . 13 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
30 eqid 2610 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵)
31 simprll 798 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
32 simpllr 795 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑗 ∈ ℤ)
33 simplrl 796 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑗))
34 simprlr 799 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
35 simprr 792 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))
3625, 28, 29, 30, 31, 32, 33, 34, 35summolem2a 14293 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → seq𝑗( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
3736expr 641 . . . . . . . . . . 11 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴) → seq𝑗( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
3837exlimdv 1848 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴) → seq𝑗( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
3924, 38mpd 15 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑗( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
40 climuni 14131 . . . . . . . . 9 ((seq𝑗( + , 𝐹) ⇝ 𝑥 ∧ seq𝑗( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
417, 39, 40syl2anc 691 . . . . . . . 8 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
4241anassrs 678 . . . . . . 7 (((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
43 eqeq2 2621 . . . . . . 7 (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦𝑥 = (seq1( + , 𝐺)‘𝑚)))
4442, 43syl5ibrcom 236 . . . . . 6 (((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
4544expimpd 627 . . . . 5 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4645exlimdv 1848 . . . 4 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4746rexlimdva 3013 . . 3 (((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4847r19.29an 3059 . 2 ((𝜑 ∧ ∃𝑗 ∈ ℤ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
496, 48sylan2b 491 1 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  wrex 2897  csb 3499  wss 3540  ifcif 4036   class class class wbr 4583  cmpt 4643   Or wor 4958  1-1-ontowf1o 5803  cfv 5804   Isom wiso 5805  (class class class)co 6549  cen 7838  Fincfn 7841  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  cn 10897  cz 11254  cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  cli 14063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067
This theorem is referenced by:  summo  14295
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