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Theorem summolem3 14378
 Description: Lemma for summo 14381. (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 9962 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 10166 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 9967 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7 summolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 475 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 11667 . . . 4 ℕ = (ℤ‘1)
108, 9syl6eleq 2708 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssid 3603 . . . 4 ℂ ⊆ ℂ
1211a1i 11 . . 3 (𝜑 → ℂ ⊆ ℂ)
13 summolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
14 f1ocnv 6106 . . . . . 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 6116 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1815, 16, 17syl2anc 692 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
19 ovex 6632 . . . . . . . . . 10 (1...𝑁) ∈ V
2019f1oen 7920 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2118, 20syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
22 fzfi 12711 . . . . . . . . 9 (1...𝑁) ∈ Fin
23 fzfi 12711 . . . . . . . . 9 (1...𝑀) ∈ Fin
24 hashen 13075 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((#‘(1...𝑁)) = (#‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2522, 23, 24mp2an 707 . . . . . . . 8 ((#‘(1...𝑁)) = (#‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2621, 25sylibr 224 . . . . . . 7 (𝜑 → (#‘(1...𝑁)) = (#‘(1...𝑀)))
277simprd 479 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
28 nnnn0 11243 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
29 hashfz1 13074 . . . . . . . 8 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
3027, 28, 293syl 18 . . . . . . 7 (𝜑 → (#‘(1...𝑁)) = 𝑁)
31 nnnn0 11243 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
32 hashfz1 13074 . . . . . . . 8 (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
338, 31, 323syl 18 . . . . . . 7 (𝜑 → (#‘(1...𝑀)) = 𝑀)
3426, 30, 333eqtr3rd 2664 . . . . . 6 (𝜑𝑀 = 𝑁)
3534oveq2d 6620 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
36 f1oeq2 6085 . . . . 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 12312 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4039adantl 482 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
41 f1of 6094 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4213, 41syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4342ffvelrnda 6315 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
44 summo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4544ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4645adantr 481 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
47 nfcsb1v 3530 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4847nfel1 2775 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
49 csbeq1a 3523 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5049eleq1d 2683 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5148, 50rspc 3289 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5243, 46, 51sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
53 fveq2 6148 . . . . . . 7 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
5453csbeq1d 3521 . . . . . 6 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
55 summo.3 . . . . . 6 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
5654, 55fvmptg 6237 . . . . 5 ((𝑚 ∈ ℕ ∧ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5740, 52, 56syl2anc 692 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5857, 52eqeltrd 2698 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
59 f1oeq2 6085 . . . . . . . . . . . 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 6094 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
64 fvco3 6232 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6563, 64sylan 488 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6665fveq2d 6152 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6713adantr 481 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6863ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
69 f1ocnvfv2 6487 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7067, 68, 69syl2anc 692 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7166, 70eqtr2d 2656 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
7271csbeq1d 3521 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
7372fveq2d 6152 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝐾𝑖) / 𝑘𝐵) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
74 elfznn 12312 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
7574adantl 482 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
76 fveq2 6148 . . . . . . 7 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
7776csbeq1d 3521 . . . . . 6 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
78 summolem3.4 . . . . . 6 𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)
7977, 78fvmpti 6238 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8075, 79syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
81 f1of 6094 . . . . . . 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 6315 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
84 elfznn 12312 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
85 fveq2 6148 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
8685csbeq1d 3521 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
8786, 55fvmpti 6238 . . . . 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 2665 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
902, 4, 6, 10, 12, 38, 58, 89seqf1o 12782 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
9134fveq2d 6152 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
9290, 91eqtr3d 2657 1 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ⦋csb 3514   ⊆ wss 3555  ifcif 4058   class class class wbr 4613   ↦ cmpt 4673   I cid 4984  ◡ccnv 5073   ∘ ccom 5078  ⟶wf 5843  –1-1-onto→wf1o 5846  ‘cfv 5847  (class class class)co 6604   ≈ cen 7896  Fincfn 7899  ℂcc 9878  0cc0 9880  1c1 9881   + caddc 9883  ℕcn 10964  ℕ0cn0 11236  ℤcz 11321  ℤ≥cuz 11631  ...cfz 12268  seqcseq 12741  #chash 13057 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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 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-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058 This theorem is referenced by:  summolem2a  14379  summo  14381
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