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Theorem summolem2 14388
Description: Lemma for summo 14389. (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 6153 . . . . 5 (𝑚 = 𝑗 → (ℤ𝑚) = (ℤ𝑗))
21sseq2d 3617 . . . 4 (𝑚 = 𝑗 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑗)))
3 seqeq1 12752 . . . . 5 (𝑚 = 𝑗 → seq𝑚( + , 𝐹) = seq𝑗( + , 𝐹))
43breq1d 4628 . . . 4 (𝑚 = 𝑗 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑗( + , 𝐹) ⇝ 𝑥))
52, 4anbi12d 746 . . 3 (𝑚 = 𝑗 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)))
65cbvrexv 3163 . 2 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑗 ∈ ℤ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥))
7 simplrr 800 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑗( + , 𝐹) ⇝ 𝑥)
8 simplrl 799 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ (ℤ𝑗))
9 uzssz 11659 . . . . . . . . . . . . . 14 (ℤ𝑗) ⊆ ℤ
10 zssre 11336 . . . . . . . . . . . . . 14 ℤ ⊆ ℝ
119, 10sstri 3596 . . . . . . . . . . . . 13 (ℤ𝑗) ⊆ ℝ
128, 11syl6ss 3599 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ ℝ)
13 ltso 10070 . . . . . . . . . . . 12 < Or ℝ
14 soss 5018 . . . . . . . . . . . 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 12719 . . . . . . . . . . . 12 (1...𝑚) ∈ Fin
17 ovex 6638 . . . . . . . . . . . . . . 15 (1...𝑚) ∈ V
1817f1oen 7928 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
1918ad2antll 764 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ≈ 𝐴)
2019ensymd 7959 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ≈ (1...𝑚))
21 enfii 8129 . . . . . . . . . . . 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 13192 . . . . . . . . . . 11 (( < Or 𝐴𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))
2415, 22, 23syl2anc 692 . . . . . . . . . 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 797 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵)
31 simprll 801 . . . . . . . . . . . . 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 798 . . . . . . . . . . . . 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 799 . . . . . . . . . . . . 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 802 . . . . . . . . . . . . 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 795 . . . . . . . . . . . . 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 14387 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 1858 . . . . . . . . . 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 14225 . . . . . . . . 9 ((seq𝑗( + , 𝐹) ⇝ 𝑥 ∧ seq𝑗( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
417, 39, 40syl2anc 692 . . . . . . . 8 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
4241anassrs 679 . . . . . . 7 (((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
43 eqeq2 2632 . . . . . . 7 (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦𝑥 = (seq1( + , 𝐺)‘𝑚)))
4442, 43syl5ibrcom 237 . . . . . 6 (((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
4544expimpd 628 . . . . 5 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4645exlimdv 1858 . . . 4 ((((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4746rexlimdva 3025 . . 3 (((𝜑𝑗 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑗) ∧ seq𝑗( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
4847r19.29an 3071 . 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 492 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  csb 3518  wss 3559  ifcif 4063   class class class wbr 4618  cmpt 4678   Or wor 4999  1-1-ontowf1o 5851  cfv 5852   Isom wiso 5853  (class class class)co 6610  cen 7904  Fincfn 7907  cc 9886  cr 9887  0cc0 9888  1c1 9889   + caddc 9891   < clt 10026  cn 10972  cz 11329  cuz 11639  ...cfz 12276  seqcseq 12749  #chash 13065  cli 14157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161
This theorem is referenced by:  summo  14389
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