Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashgadd Structured version   Visualization version   GIF version

 Description: 𝐺 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
Hypothesis
Ref Expression
hashgadd.1 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
Assertion
Ref Expression
hashgadd ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))

Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . . 6 (𝑛 = ∅ → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 ∅))
21fveq2d 6107 . . . . 5 (𝑛 = ∅ → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 ∅)))
3 fveq2 6103 . . . . . 6 (𝑛 = ∅ → (𝐺𝑛) = (𝐺‘∅))
43oveq2d 6565 . . . . 5 (𝑛 = ∅ → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺‘∅)))
52, 4eqeq12d 2625 . . . 4 (𝑛 = ∅ → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅))))
65imbi2d 329 . . 3 (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅)))))
7 oveq2 6557 . . . . . 6 (𝑛 = 𝑧 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 𝑧))
87fveq2d 6107 . . . . 5 (𝑛 = 𝑧 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 𝑧)))
9 fveq2 6103 . . . . . 6 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
109oveq2d 6565 . . . . 5 (𝑛 = 𝑧 → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺𝑧)))
118, 10eqeq12d 2625 . . . 4 (𝑛 = 𝑧 → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))))
1211imbi2d 329 . . 3 (𝑛 = 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)))))
13 oveq2 6557 . . . . . 6 (𝑛 = suc 𝑧 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 suc 𝑧))
1413fveq2d 6107 . . . . 5 (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 suc 𝑧)))
15 fveq2 6103 . . . . . 6 (𝑛 = suc 𝑧 → (𝐺𝑛) = (𝐺‘suc 𝑧))
1615oveq2d 6565 . . . . 5 (𝑛 = suc 𝑧 → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))
1714, 16eqeq12d 2625 . . . 4 (𝑛 = suc 𝑧 → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧))))
1817imbi2d 329 . . 3 (𝑛 = suc 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))))
19 oveq2 6557 . . . . . 6 (𝑛 = 𝐵 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 𝐵))
2019fveq2d 6107 . . . . 5 (𝑛 = 𝐵 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 𝐵)))
21 fveq2 6103 . . . . . 6 (𝑛 = 𝐵 → (𝐺𝑛) = (𝐺𝐵))
2221oveq2d 6565 . . . . 5 (𝑛 = 𝐵 → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺𝐵)))
2320, 22eqeq12d 2625 . . . 4 (𝑛 = 𝐵 → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵))))
2423imbi2d 329 . . 3 (𝑛 = 𝐵 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))))
25 hashgadd.1 . . . . . . . . 9 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2625hashgf1o 12632 . . . . . . . 8 𝐺:ω–1-1-onto→ℕ0
27 f1of 6050 . . . . . . . 8 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
2826, 27ax-mp 5 . . . . . . 7 𝐺:ω⟶ℕ0
2928ffvelrni 6266 . . . . . 6 (𝐴 ∈ ω → (𝐺𝐴) ∈ ℕ0)
3029nn0cnd 11230 . . . . 5 (𝐴 ∈ ω → (𝐺𝐴) ∈ ℂ)
3130addid1d 10115 . . . 4 (𝐴 ∈ ω → ((𝐺𝐴) + 0) = (𝐺𝐴))
32 0z 11265 . . . . . . 7 0 ∈ ℤ
3332, 25om2uz0i 12608 . . . . . 6 (𝐺‘∅) = 0
3433oveq2i 6560 . . . . 5 ((𝐺𝐴) + (𝐺‘∅)) = ((𝐺𝐴) + 0)
3534a1i 11 . . . 4 (𝐴 ∈ ω → ((𝐺𝐴) + (𝐺‘∅)) = ((𝐺𝐴) + 0))
36 nna0 7571 . . . . 5 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
3736fveq2d 6107 . . . 4 (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = (𝐺𝐴))
3831, 35, 373eqtr4rd 2655 . . 3 (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅)))
39 nnasuc 7573 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
4039fveq2d 6107 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = (𝐺‘suc (𝐴 +𝑜 𝑧)))
41 nnacl 7578 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +𝑜 𝑧) ∈ ω)
4232, 25om2uzsuci 12609 . . . . . . . . . 10 ((𝐴 +𝑜 𝑧) ∈ ω → (𝐺‘suc (𝐴 +𝑜 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
4341, 42syl 17 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘suc (𝐴 +𝑜 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
4440, 43eqtrd 2644 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
45443adant3 1074 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
4628ffvelrni 6266 . . . . . . . . . . 11 (𝑧 ∈ ω → (𝐺𝑧) ∈ ℕ0)
4746nn0cnd 11230 . . . . . . . . . 10 (𝑧 ∈ ω → (𝐺𝑧) ∈ ℂ)
48 ax-1cn 9873 . . . . . . . . . . 11 1 ∈ ℂ
49 addass 9902 . . . . . . . . . . 11 (((𝐺𝐴) ∈ ℂ ∧ (𝐺𝑧) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺𝐴) + (𝐺𝑧)) + 1) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
5048, 49mp3an3 1405 . . . . . . . . . 10 (((𝐺𝐴) ∈ ℂ ∧ (𝐺𝑧) ∈ ℂ) → (((𝐺𝐴) + (𝐺𝑧)) + 1) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
5130, 47, 50syl2an 493 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐺𝐴) + (𝐺𝑧)) + 1) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
52513adant3 1074 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (((𝐺𝐴) + (𝐺𝑧)) + 1) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
53 oveq1 6556 . . . . . . . . 9 ((𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)) → ((𝐺‘(𝐴 +𝑜 𝑧)) + 1) = (((𝐺𝐴) + (𝐺𝑧)) + 1))
54533ad2ant3 1077 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → ((𝐺‘(𝐴 +𝑜 𝑧)) + 1) = (((𝐺𝐴) + (𝐺𝑧)) + 1))
5532, 25om2uzsuci 12609 . . . . . . . . . 10 (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺𝑧) + 1))
5655oveq2d 6565 . . . . . . . . 9 (𝑧 ∈ ω → ((𝐺𝐴) + (𝐺‘suc 𝑧)) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
57563ad2ant2 1076 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → ((𝐺𝐴) + (𝐺‘suc 𝑧)) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
5852, 54, 573eqtr4d 2654 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → ((𝐺‘(𝐴 +𝑜 𝑧)) + 1) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))
5945, 58eqtrd 2644 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))
60593expia 1259 . . . . 5 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧))))
6160expcom 450 . . . 4 (𝑧 ∈ ω → (𝐴 ∈ ω → ((𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))))
6261a2d 29 . . 3 (𝑧 ∈ ω → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))))
636, 12, 18, 24, 38, 62finds 6984 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵))))
6463impcom 445 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   ↦ cmpt 4643   ↾ cres 5040  suc csuc 5642  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392   +𝑜 coa 7444  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  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 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-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-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-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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564 This theorem is referenced by:  hashdom  13029  hashun  13032
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