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Theorem summolem3 14653
Description: Lemma for summo 14656. (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 10224 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 467 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 10428 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 467 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 10229 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 467 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7 summolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 482 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 11930 . . . 4 ℕ = (ℤ‘1)
108, 9syl6eleq 2860 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssid 3773 . . . 4 ℂ ⊆ ℂ
1211a1i 11 . . 3 (𝜑 → ℂ ⊆ ℂ)
13 summolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
14 f1ocnv 6291 . . . . . 6 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:𝐴1-1-onto→(1...𝑀))
1513, 14syl 17 . . . . 5 (𝜑𝑓:𝐴1-1-onto→(1...𝑀))
16 summolem3.7 . . . . 5 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
17 f1oco 6301 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1815, 16, 17syl2anc 573 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
19 ovex 6827 . . . . . . . . . 10 (1...𝑁) ∈ V
2019f1oen 8134 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2118, 20syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
22 fzfi 12979 . . . . . . . . 9 (1...𝑁) ∈ Fin
23 fzfi 12979 . . . . . . . . 9 (1...𝑀) ∈ Fin
24 hashen 13339 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2522, 23, 24mp2an 672 . . . . . . . 8 ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2621, 25sylibr 224 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
277simprd 483 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
28 nnnn0 11506 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
29 hashfz1 13338 . . . . . . . 8 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
3027, 28, 293syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
31 nnnn0 11506 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
32 hashfz1 13338 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
338, 31, 323syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
3426, 30, 333eqtr3rd 2814 . . . . . 6 (𝜑𝑀 = 𝑁)
3534oveq2d 6812 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
36 f1oeq2 6270 . . . . 5 ((1...𝑀) = (1...𝑁) → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3735, 36syl 17 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3818, 37mpbird 247 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
39 elfznn 12577 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4039adantl 467 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
41 f1of 6279 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4213, 41syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4342ffvelrnda 6504 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
44 summo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4544ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4645adantr 466 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
47 nfcsb1v 3698 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4847nfel1 2928 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
49 csbeq1a 3691 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5049eleq1d 2835 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5148, 50rspc 3454 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5243, 46, 51sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
53 fveq2 6333 . . . . . . 7 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
5453csbeq1d 3689 . . . . . 6 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
55 summo.3 . . . . . 6 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
5654, 55fvmptg 6424 . . . . 5 ((𝑚 ∈ ℕ ∧ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5740, 52, 56syl2anc 573 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5857, 52eqeltrd 2850 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
59 f1oeq2 6270 . . . . . . . . . . . 12 ((1...𝑀) = (1...𝑁) → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
6035, 59syl 17 . . . . . . . . . . 11 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
6116, 60mpbird 247 . . . . . . . . . 10 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
62 f1of 6279 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
64 fvco3 6419 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6563, 64sylan 569 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6665fveq2d 6337 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6713adantr 466 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6863ffvelrnda 6504 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
69 f1ocnvfv2 6679 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7067, 68, 69syl2anc 573 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7166, 70eqtr2d 2806 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
7271csbeq1d 3689 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
7372fveq2d 6337 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝐾𝑖) / 𝑘𝐵) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
74 elfznn 12577 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
7574adantl 467 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
76 fveq2 6333 . . . . . . 7 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
7776csbeq1d 3689 . . . . . 6 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
78 summolem3.4 . . . . . 6 𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)
7977, 78fvmpti 6425 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8075, 79syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
81 f1of 6279 . . . . . . 7 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
8238, 81syl 17 . . . . . 6 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
8382ffvelrnda 6504 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
84 elfznn 12577 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
85 fveq2 6333 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
8685csbeq1d 3689 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
8786, 55fvmpti 6425 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ ℕ → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8883, 84, 873syl 18 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8973, 80, 883eqtr4d 2815 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
902, 4, 6, 10, 12, 38, 58, 89seqf1o 13049 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
9134fveq2d 6337 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
9290, 91eqtr3d 2807 1 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  csb 3682  wss 3723  ifcif 4226   class class class wbr 4787  cmpt 4864   I cid 5157  ccnv 5249  ccom 5254  wf 6026  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  cen 8110  Fincfn 8113  cc 10140  0cc0 10142  1c1 10143   + caddc 10145  cn 11226  0cn0 11499  cz 11584  cuz 11893  ...cfz 12533  seqcseq 13008  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-n0 11500  df-z 11585  df-uz 11894  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322
This theorem is referenced by:  summolem2a  14654  summo  14656
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