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Theorem omgadd 9878
Description: Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
Hypothesis
Ref Expression
omgadd.1 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
Assertion
Ref Expression
omgadd ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))

Proof of Theorem omgadd
Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5571 . . . . . 6 (𝑛 = ∅ → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 ∅))
21fveq2d 5233 . . . . 5 (𝑛 = ∅ → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 ∅)))
3 fveq2 5229 . . . . . 6 (𝑛 = ∅ → (𝐺𝑛) = (𝐺‘∅))
43oveq2d 5579 . . . . 5 (𝑛 = ∅ → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺‘∅)))
52, 4eqeq12d 2097 . . . 4 (𝑛 = ∅ → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅))))
65imbi2d 228 . . 3 (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅)))))
7 oveq2 5571 . . . . . 6 (𝑛 = 𝑧 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 𝑧))
87fveq2d 5233 . . . . 5 (𝑛 = 𝑧 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 𝑧)))
9 fveq2 5229 . . . . . 6 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
109oveq2d 5579 . . . . 5 (𝑛 = 𝑧 → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺𝑧)))
118, 10eqeq12d 2097 . . . 4 (𝑛 = 𝑧 → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))))
1211imbi2d 228 . . 3 (𝑛 = 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)))))
13 oveq2 5571 . . . . . 6 (𝑛 = suc 𝑧 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 suc 𝑧))
1413fveq2d 5233 . . . . 5 (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 suc 𝑧)))
15 fveq2 5229 . . . . . 6 (𝑛 = suc 𝑧 → (𝐺𝑛) = (𝐺‘suc 𝑧))
1615oveq2d 5579 . . . . 5 (𝑛 = suc 𝑧 → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))
1714, 16eqeq12d 2097 . . . 4 (𝑛 = suc 𝑧 → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧))))
1817imbi2d 228 . . 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 5571 . . . . . 6 (𝑛 = 𝐵 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 𝐵))
2019fveq2d 5233 . . . . 5 (𝑛 = 𝐵 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 𝐵)))
21 fveq2 5229 . . . . . 6 (𝑛 = 𝐵 → (𝐺𝑛) = (𝐺𝐵))
2221oveq2d 5579 . . . . 5 (𝑛 = 𝐵 → ((𝐺𝐴) + (𝐺𝑛)) = ((𝐺𝐴) + (𝐺𝐵)))
2320, 22eqeq12d 2097 . . . 4 (𝑛 = 𝐵 → ((𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛)) ↔ (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵))))
2423imbi2d 228 . . 3 (𝑛 = 𝐵 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝑛)) = ((𝐺𝐴) + (𝐺𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))))
25 omgadd.1 . . . . . . . . 9 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
2625frechashgf1o 9562 . . . . . . . 8 𝐺:ω–1-1-onto→ℕ0
27 f1of 5177 . . . . . . . 8 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
2826, 27ax-mp 7 . . . . . . 7 𝐺:ω⟶ℕ0
2928ffvelrni 5353 . . . . . 6 (𝐴 ∈ ω → (𝐺𝐴) ∈ ℕ0)
3029nn0cnd 8462 . . . . 5 (𝐴 ∈ ω → (𝐺𝐴) ∈ ℂ)
3130addid1d 7376 . . . 4 (𝐴 ∈ ω → ((𝐺𝐴) + 0) = (𝐺𝐴))
32 0zd 8496 . . . . . 6 (𝐴 ∈ ω → 0 ∈ ℤ)
3332, 25frec2uz0d 9533 . . . . 5 (𝐴 ∈ ω → (𝐺‘∅) = 0)
3433oveq2d 5579 . . . 4 (𝐴 ∈ ω → ((𝐺𝐴) + (𝐺‘∅)) = ((𝐺𝐴) + 0))
35 nna0 6138 . . . . 5 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
3635fveq2d 5233 . . . 4 (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = (𝐺𝐴))
3731, 34, 363eqtr4rd 2126 . . 3 (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅)))
38 nnasuc 6140 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
3938fveq2d 5233 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = (𝐺‘suc (𝐴 +𝑜 𝑧)))
40 0zd 8496 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → 0 ∈ ℤ)
41 nnacl 6144 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +𝑜 𝑧) ∈ ω)
4240, 25, 41frec2uzsucd 9535 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘suc (𝐴 +𝑜 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
4339, 42eqtrd 2115 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
44433adant3 959 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺‘(𝐴 +𝑜 𝑧)) + 1))
45303ad2ant1 960 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺𝐴) ∈ ℂ)
4628ffvelrni 5353 . . . . . . . . . . 11 (𝑧 ∈ ω → (𝐺𝑧) ∈ ℕ0)
4746nn0cnd 8462 . . . . . . . . . 10 (𝑧 ∈ ω → (𝐺𝑧) ∈ ℂ)
48473ad2ant2 961 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺𝑧) ∈ ℂ)
49 1cnd 7249 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → 1 ∈ ℂ)
5045, 48, 49addassd 7255 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (((𝐺𝐴) + (𝐺𝑧)) + 1) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
51 oveq1 5570 . . . . . . . . 9 ((𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)) → ((𝐺‘(𝐴 +𝑜 𝑧)) + 1) = (((𝐺𝐴) + (𝐺𝑧)) + 1))
52513ad2ant3 962 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → ((𝐺‘(𝐴 +𝑜 𝑧)) + 1) = (((𝐺𝐴) + (𝐺𝑧)) + 1))
53 0zd 8496 . . . . . . . . . . 11 (𝑧 ∈ ω → 0 ∈ ℤ)
54 id 19 . . . . . . . . . . 11 (𝑧 ∈ ω → 𝑧 ∈ ω)
5553, 25, 54frec2uzsucd 9535 . . . . . . . . . 10 (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺𝑧) + 1))
5655oveq2d 5579 . . . . . . . . 9 (𝑧 ∈ ω → ((𝐺𝐴) + (𝐺‘suc 𝑧)) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
57563ad2ant2 961 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → ((𝐺𝐴) + (𝐺‘suc 𝑧)) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
5850, 52, 573eqtr4d 2125 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → ((𝐺‘(𝐴 +𝑜 𝑧)) + 1) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))
5944, 58eqtrd 2115 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))
60593expia 1141 . . . . 5 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧))))
6160expcom 114 . . . 4 (𝑧 ∈ ω → (𝐴 ∈ ω → ((𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧)) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = ((𝐺𝐴) + (𝐺‘suc 𝑧)))))
6261a2d 26 . . 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, 37, 62finds 4369 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵))))
6463impcom 123 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  c0 3267  cmpt 3859  suc csuc 4148  ωcom 4359  wf 4948  1-1-ontowf1o 4951  cfv 4952  (class class class)co 5563  freccfrec 6059   +𝑜 coa 6082  cc 7093  0cc0 7095  1c1 7096   + caddc 7098  0cn0 8407  cz 8484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-frec 6060  df-oadd 6089  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753
This theorem is referenced by:  hashun  9881
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