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Theorem summolem3 14786
Description: Lemma for summo 14789. (Contributed by Mario Carneiro, 29-Mar-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)
summolem3.4 𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)
summolem3.5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
summolem3.6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
summolem3.7 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
Assertion
Ref Expression
summolem3 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Distinct variable groups:   𝑓,𝑘,𝑛,𝐴   𝑓,𝐹,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝐾,𝑛   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝐵,𝑓,𝑛   𝑘,𝑀,𝑛
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑘)   𝐺(𝑓)   𝐻(𝑓,𝑘,𝑛)   𝐾(𝑓)   𝑀(𝑓)   𝑁(𝑓)

Proof of Theorem summolem3
Dummy variables 𝑖 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 10306 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 474 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 10512 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 474 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 10311 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 474 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7 summolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 489 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 11967 . . . 4 ℕ = (ℤ‘1)
108, 9syl6eleq 2888 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssidd 3820 . . 3 (𝜑 → ℂ ⊆ ℂ)
12 summolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
13 f1ocnv 6368 . . . . . 6 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:𝐴1-1-onto→(1...𝑀))
1412, 13syl 17 . . . . 5 (𝜑𝑓:𝐴1-1-onto→(1...𝑀))
15 summolem3.7 . . . . 5 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
16 f1oco 6378 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1714, 15, 16syl2anc 580 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18 ovex 6910 . . . . . . . . . 10 (1...𝑁) ∈ V
1918f1oen 8216 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2017, 19syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
21 fzfi 13026 . . . . . . . . 9 (1...𝑁) ∈ Fin
22 fzfi 13026 . . . . . . . . 9 (1...𝑀) ∈ Fin
23 hashen 13387 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2421, 22, 23mp2an 684 . . . . . . . 8 ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2520, 24sylibr 226 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
267simprd 490 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
27 nnnn0 11588 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
28 hashfz1 13386 . . . . . . . 8 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2926, 27, 283syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30 nnnn0 11588 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
31 hashfz1 13386 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
328, 30, 313syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
3325, 29, 323eqtr3rd 2842 . . . . . 6 (𝜑𝑀 = 𝑁)
3433oveq2d 6894 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
3534f1oeq2d 6352 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3617, 35mpbird 249 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37 elfznn 12624 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
3837adantl 474 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
39 f1of 6356 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4012, 39syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4140ffvelrnda 6585 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
42 summo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4342ralrimiva 3147 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4443adantr 473 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
45 nfcsb1v 3744 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4645nfel1 2956 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
47 csbeq1a 3737 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
4847eleq1d 2863 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4946, 48rspc 3491 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5041, 44, 49sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
51 fveq2 6411 . . . . . . 7 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
5251csbeq1d 3735 . . . . . 6 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
53 summo.3 . . . . . 6 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
5452, 53fvmptg 6505 . . . . 5 ((𝑚 ∈ ℕ ∧ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5538, 50, 54syl2anc 580 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5655, 50eqeltrd 2878 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
5734f1oeq2d 6352 . . . . . . . . . . 11 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
5815, 57mpbird 249 . . . . . . . . . 10 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
59 f1of 6356 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
6058, 59syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
61 fvco3 6500 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6260, 61sylan 576 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6362fveq2d 6415 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6412adantr 473 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6560ffvelrnda 6585 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
66 f1ocnvfv2 6761 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
6764, 65, 66syl2anc 580 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
6863, 67eqtr2d 2834 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
6968csbeq1d 3735 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
7069fveq2d 6415 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝐾𝑖) / 𝑘𝐵) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
71 elfznn 12624 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
7271adantl 474 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
73 fveq2 6411 . . . . . . 7 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
7473csbeq1d 3735 . . . . . 6 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
75 summolem3.4 . . . . . 6 𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)
7674, 75fvmpti 6506 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
7772, 76syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
78 f1of 6356 . . . . . . 7 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7936, 78syl 17 . . . . . 6 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
8079ffvelrnda 6585 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
81 elfznn 12624 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
82 fveq2 6411 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
8382csbeq1d 3735 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
8483, 53fvmpti 6506 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ ℕ → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8580, 81, 843syl 18 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8670, 77, 853eqtr4d 2843 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
872, 4, 6, 10, 11, 36, 56, 86seqf1o 13096 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
8833fveq2d 6415 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
8987, 88eqtr3d 2835 1 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  csb 3728  ifcif 4277   class class class wbr 4843  cmpt 4922   I cid 5219  ccnv 5311  ccom 5316  wf 6097  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  cen 8192  Fincfn 8195  cc 10222  0cc0 10224  1c1 10225   + caddc 10227  cn 11312  0cn0 11580  cz 11666  cuz 11930  ...cfz 12580  seqcseq 13055  chash 13370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-seq 13056  df-hash 13371
This theorem is referenced by:  summolem2a  14787  summo  14789
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