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Theorem onov0suclim 42327
Description: Compactly express rules for binary operations on ordinals. (Contributed by RP, 18-Jan-2025.)
Hypotheses
Ref Expression
onov0suclim.0 (𝐴 ∈ On → (𝐴 ∅) = 𝐷)
onov0suclim.suc ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 suc 𝐶) = 𝐸)
onov0suclim.lim (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 𝐵) = 𝐹)
Assertion
Ref Expression
onov0suclim ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹)))

Proof of Theorem onov0suclim
StepHypRef Expression
1 eloni 6374 . . . 4 (𝐵 ∈ On → Ord 𝐵)
2 orduniorsuc 7822 . . . . 5 (Ord 𝐵 → (𝐵 = 𝐵𝐵 = suc 𝐵))
3 unizlim 6487 . . . . . . 7 (Ord 𝐵 → (𝐵 = 𝐵 ↔ (𝐵 = ∅ ∨ Lim 𝐵)))
43biimpd 228 . . . . . 6 (Ord 𝐵 → (𝐵 = 𝐵 → (𝐵 = ∅ ∨ Lim 𝐵)))
54orim1d 963 . . . . 5 (Ord 𝐵 → ((𝐵 = 𝐵𝐵 = suc 𝐵) → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc 𝐵)))
62, 5mpd 15 . . . 4 (Ord 𝐵 → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc 𝐵))
71, 6syl 17 . . 3 (𝐵 ∈ On → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc 𝐵))
87adantl 481 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc 𝐵))
9 oveq2 7420 . . . . . . . . 9 (𝐵 = ∅ → (𝐴 𝐵) = (𝐴 ∅))
10 onov0suclim.0 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 ∅) = 𝐷)
119, 10sylan9eqr 2793 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 𝐵) = 𝐷)
1211ex 412 . . . . . . 7 (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 𝐵) = 𝐷))
1312ad2antrr 723 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → (𝐵 = ∅ → (𝐴 𝐵) = 𝐷))
14 eloni 6374 . . . . . . . . . . . . 13 (𝐶 ∈ On → Ord 𝐶)
15 0elsuc 7827 . . . . . . . . . . . . 13 (Ord 𝐶 → ∅ ∈ suc 𝐶)
1614, 15syl 17 . . . . . . . . . . . 12 (𝐶 ∈ On → ∅ ∈ suc 𝐶)
1716adantl 481 . . . . . . . . . . 11 ((𝐵 = suc 𝐶𝐶 ∈ On) → ∅ ∈ suc 𝐶)
18 simpl 482 . . . . . . . . . . 11 ((𝐵 = suc 𝐶𝐶 ∈ On) → 𝐵 = suc 𝐶)
1917, 18eleqtrrd 2835 . . . . . . . . . 10 ((𝐵 = suc 𝐶𝐶 ∈ On) → ∅ ∈ 𝐵)
20 n0i 4333 . . . . . . . . . 10 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
2119, 20syl 17 . . . . . . . . 9 ((𝐵 = suc 𝐶𝐶 ∈ On) → ¬ 𝐵 = ∅)
2221pm2.21d 121 . . . . . . . 8 ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐵 = ∅ → (𝐴 𝐵) = 𝐸))
2322adantl 481 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → (𝐵 = ∅ → (𝐴 𝐵) = 𝐸))
2423impancom 451 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸))
25 nlim0 6423 . . . . . . . . 9 ¬ Lim ∅
26 limeq 6376 . . . . . . . . 9 (𝐵 = ∅ → (Lim 𝐵 ↔ Lim ∅))
2725, 26mtbiri 327 . . . . . . . 8 (𝐵 = ∅ → ¬ Lim 𝐵)
2827adantl 481 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → ¬ Lim 𝐵)
2928pm2.21d 121 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → (Lim 𝐵 → (𝐴 𝐵) = 𝐹))
3013, 24, 293jca 1127 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹)))
3130ex 412 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹))))
3227con2i 139 . . . . . . . 8 (Lim 𝐵 → ¬ 𝐵 = ∅)
3332adantl 481 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → ¬ 𝐵 = ∅)
3433pm2.21d 121 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐵 = ∅ → (𝐴 𝐵) = 𝐷))
35 limeq 6376 . . . . . . . . . . . 12 (𝐵 = suc 𝐶 → (Lim 𝐵 ↔ Lim suc 𝐶))
3635notbid 318 . . . . . . . . . . 11 (𝐵 = suc 𝐶 → (¬ Lim 𝐵 ↔ ¬ Lim suc 𝐶))
3736biimprd 247 . . . . . . . . . 10 (𝐵 = suc 𝐶 → (¬ Lim suc 𝐶 → ¬ Lim 𝐵))
38 nlimsucg 7835 . . . . . . . . . 10 (𝐶 ∈ On → ¬ Lim suc 𝐶)
3937, 38impel 505 . . . . . . . . 9 ((𝐵 = suc 𝐶𝐶 ∈ On) → ¬ Lim 𝐵)
4039adantl 481 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → ¬ Lim 𝐵)
4140pm2.21d 121 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → (Lim 𝐵 → (𝐴 𝐵) = 𝐸))
4241impancom 451 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸))
43 onov0suclim.lim . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 𝐵) = 𝐹)
4443a1d 25 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (Lim 𝐵 → (𝐴 𝐵) = 𝐹))
4534, 42, 443jca 1127 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹)))
4645ex 412 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Lim 𝐵 → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹))))
4731, 46jaod 856 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ ∨ Lim 𝐵) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹))))
48 1n0 8492 . . . . . . . . 9 1o ≠ ∅
49 necom 2993 . . . . . . . . . . 11 (1o ≠ ∅ ↔ ∅ ≠ 1o)
50 df-1o 8470 . . . . . . . . . . . . 13 1o = suc ∅
51 uni0 4939 . . . . . . . . . . . . . 14 ∅ = ∅
52 suceq 6430 . . . . . . . . . . . . . 14 ( ∅ = ∅ → suc ∅ = suc ∅)
5351, 52ax-mp 5 . . . . . . . . . . . . 13 suc ∅ = suc ∅
5450, 53eqtr4i 2762 . . . . . . . . . . . 12 1o = suc
5554neeq2i 3005 . . . . . . . . . . 11 (∅ ≠ 1o ↔ ∅ ≠ suc ∅)
56 df-ne 2940 . . . . . . . . . . 11 (∅ ≠ suc ∅ ↔ ¬ ∅ = suc ∅)
5749, 55, 563bitri 297 . . . . . . . . . 10 (1o ≠ ∅ ↔ ¬ ∅ = suc ∅)
58 id 22 . . . . . . . . . . . 12 (𝐵 = ∅ → 𝐵 = ∅)
59 unieq 4919 . . . . . . . . . . . . 13 (𝐵 = ∅ → 𝐵 = ∅)
60 suceq 6430 . . . . . . . . . . . . 13 ( 𝐵 = ∅ → suc 𝐵 = suc ∅)
6159, 60syl 17 . . . . . . . . . . . 12 (𝐵 = ∅ → suc 𝐵 = suc ∅)
6258, 61eqeq12d 2747 . . . . . . . . . . 11 (𝐵 = ∅ → (𝐵 = suc 𝐵 ↔ ∅ = suc ∅))
6362notbid 318 . . . . . . . . . 10 (𝐵 = ∅ → (¬ 𝐵 = suc 𝐵 ↔ ¬ ∅ = suc ∅))
6457, 63bitr4id 290 . . . . . . . . 9 (𝐵 = ∅ → (1o ≠ ∅ ↔ ¬ 𝐵 = suc 𝐵))
6548, 64mpbii 232 . . . . . . . 8 (𝐵 = ∅ → ¬ 𝐵 = suc 𝐵)
6665con2i 139 . . . . . . 7 (𝐵 = suc 𝐵 → ¬ 𝐵 = ∅)
6766adantl 481 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc 𝐵) → ¬ 𝐵 = ∅)
6867pm2.21d 121 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc 𝐵) → (𝐵 = ∅ → (𝐴 𝐵) = 𝐷))
69 simprl 768 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → 𝐵 = suc 𝐶)
7069oveq2d 7428 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → (𝐴 𝐵) = (𝐴 suc 𝐶))
71 onov0suclim.suc . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 suc 𝐶) = 𝐸)
7271adantrl 713 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → (𝐴 suc 𝐶) = 𝐸)
7370, 72eqtrd 2771 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶𝐶 ∈ On)) → (𝐴 𝐵) = 𝐸)
7473ex 412 . . . . . 6 (𝐴 ∈ On → ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸))
7574ad2antrr 723 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc 𝐵) → ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸))
76 onuni 7780 . . . . . . . . 9 (𝐵 ∈ On → 𝐵 ∈ On)
77 nlimsucg 7835 . . . . . . . . 9 ( 𝐵 ∈ On → ¬ Lim suc 𝐵)
7876, 77syl 17 . . . . . . . 8 (𝐵 ∈ On → ¬ Lim suc 𝐵)
79 limeq 6376 . . . . . . . . . 10 (𝐵 = suc 𝐵 → (Lim 𝐵 ↔ Lim suc 𝐵))
8079notbid 318 . . . . . . . . 9 (𝐵 = suc 𝐵 → (¬ Lim 𝐵 ↔ ¬ Lim suc 𝐵))
8180biimprd 247 . . . . . . . 8 (𝐵 = suc 𝐵 → (¬ Lim suc 𝐵 → ¬ Lim 𝐵))
8278, 81mpan9 506 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵 = suc 𝐵) → ¬ Lim 𝐵)
8382adantll 711 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc 𝐵) → ¬ Lim 𝐵)
8483pm2.21d 121 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc 𝐵) → (Lim 𝐵 → (𝐴 𝐵) = 𝐹))
8568, 75, 843jca 1127 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc 𝐵) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹)))
8685ex 412 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = suc 𝐵 → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹))))
8747, 86jaod 856 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc 𝐵) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹))))
888, 87mpd 15 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶𝐶 ∈ On) → (𝐴 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 𝐵) = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  wne 2939  c0 4322   cuni 4908  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7412  1oc1o 8463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fv 6551  df-ov 7415  df-1o 8470
This theorem is referenced by:  oa0suclim  42328  om0suclim  42329  oe0suclim  42330
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