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Theorem omgadd 10043
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 5597 . . . . . 6 (𝑛 = ∅ → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 ∅))
21fveq2d 5255 . . . . 5 (𝑛 = ∅ → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 ∅)))
3 fveq2 5251 . . . . . 6 (𝑛 = ∅ → (𝐺𝑛) = (𝐺‘∅))
43oveq2d 5605 . . . . 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 5597 . . . . . 6 (𝑛 = 𝑧 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 𝑧))
87fveq2d 5255 . . . . 5 (𝑛 = 𝑧 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 𝑧)))
9 fveq2 5251 . . . . . 6 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
109oveq2d 5605 . . . . 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 5597 . . . . . 6 (𝑛 = suc 𝑧 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 suc 𝑧))
1413fveq2d 5255 . . . . 5 (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 suc 𝑧)))
15 fveq2 5251 . . . . . 6 (𝑛 = suc 𝑧 → (𝐺𝑛) = (𝐺‘suc 𝑧))
1615oveq2d 5605 . . . . 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 5597 . . . . . 6 (𝑛 = 𝐵 → (𝐴 +𝑜 𝑛) = (𝐴 +𝑜 𝐵))
2019fveq2d 5255 . . . . 5 (𝑛 = 𝐵 → (𝐺‘(𝐴 +𝑜 𝑛)) = (𝐺‘(𝐴 +𝑜 𝐵)))
21 fveq2 5251 . . . . . 6 (𝑛 = 𝐵 → (𝐺𝑛) = (𝐺𝐵))
2221oveq2d 5605 . . . . 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 9722 . . . . . . . 8 𝐺:ω–1-1-onto→ℕ0
27 f1of 5199 . . . . . . . 8 (𝐺:ω–1-1-onto→ℕ0𝐺:ω⟶ℕ0)
2826, 27ax-mp 7 . . . . . . 7 𝐺:ω⟶ℕ0
2928ffvelrni 5376 . . . . . 6 (𝐴 ∈ ω → (𝐺𝐴) ∈ ℕ0)
3029nn0cnd 8618 . . . . 5 (𝐴 ∈ ω → (𝐺𝐴) ∈ ℂ)
3130addid1d 7532 . . . 4 (𝐴 ∈ ω → ((𝐺𝐴) + 0) = (𝐺𝐴))
32 0zd 8656 . . . . . 6 (𝐴 ∈ ω → 0 ∈ ℤ)
3332, 25frec2uz0d 9693 . . . . 5 (𝐴 ∈ ω → (𝐺‘∅) = 0)
3433oveq2d 5605 . . . 4 (𝐴 ∈ ω → ((𝐺𝐴) + (𝐺‘∅)) = ((𝐺𝐴) + 0))
35 nna0 6165 . . . . 5 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
3635fveq2d 5255 . . . 4 (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = (𝐺𝐴))
3731, 34, 363eqtr4rd 2126 . . 3 (𝐴 ∈ ω → (𝐺‘(𝐴 +𝑜 ∅)) = ((𝐺𝐴) + (𝐺‘∅)))
38 nnasuc 6167 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
3938fveq2d 5255 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +𝑜 suc 𝑧)) = (𝐺‘suc (𝐴 +𝑜 𝑧)))
40 0zd 8656 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → 0 ∈ ℤ)
41 nnacl 6171 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +𝑜 𝑧) ∈ ω)
4240, 25, 41frec2uzsucd 9695 . . . . . . . . 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 5376 . . . . . . . . . . 11 (𝑧 ∈ ω → (𝐺𝑧) ∈ ℕ0)
4746nn0cnd 8618 . . . . . . . . . 10 (𝑧 ∈ ω → (𝐺𝑧) ∈ ℂ)
48473ad2ant2 961 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (𝐺𝑧) ∈ ℂ)
49 1cnd 7405 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → 1 ∈ ℂ)
5045, 48, 49addassd 7411 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +𝑜 𝑧)) = ((𝐺𝐴) + (𝐺𝑧))) → (((𝐺𝐴) + (𝐺𝑧)) + 1) = ((𝐺𝐴) + ((𝐺𝑧) + 1)))
51 oveq1 5596 . . . . . . . . 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 8656 . . . . . . . . . . 11 (𝑧 ∈ ω → 0 ∈ ℤ)
54 id 19 . . . . . . . . . . 11 (𝑧 ∈ ω → 𝑧 ∈ ω)
5553, 25, 54frec2uzsucd 9695 . . . . . . . . . 10 (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺𝑧) + 1))
5655oveq2d 5605 . . . . . . . . 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 4377 . 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 3269  cmpt 3865  suc csuc 4155  ωcom 4367  wf 4963  1-1-ontowf1o 4966  cfv 4967  (class class class)co 5589  freccfrec 6085   +𝑜 coa 6108  cc 7249  0cc0 7251  1c1 7252   + caddc 7254  0cn0 8563  cz 8644
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 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-addcom 7346  ax-addass 7348  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-0id 7354  ax-rnegex 7355  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-ltadd 7362
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-frec 6086  df-oadd 6115  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-inn 8315  df-n0 8564  df-z 8645  df-uz 8913
This theorem is referenced by:  hashun  10046
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