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

Theorem noextendseq 27160
Description: Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.)
Hypothesis
Ref Expression
noextend.1 𝑋 ∈ {1o, 2o}
Assertion
Ref Expression
noextendseq ((𝐴 No 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )

Proof of Theorem noextendseq
StepHypRef Expression
1 nofun 27142 . . . 4 (𝐴 No → Fun 𝐴)
2 noextend.1 . . . . 5 𝑋 ∈ {1o, 2o}
3 fnconstg 6777 . . . . 5 (𝑋 ∈ {1o, 2o} → ((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴))
4 fnfun 6647 . . . . 5 (((𝐵 ∖ dom 𝐴) × {𝑋}) Fn (𝐵 ∖ dom 𝐴) → Fun ((𝐵 ∖ dom 𝐴) × {𝑋}))
52, 3, 4mp2b 10 . . . 4 Fun ((𝐵 ∖ dom 𝐴) × {𝑋})
6 snnzg 4778 . . . . . . . 8 (𝑋 ∈ {1o, 2o} → {𝑋} ≠ ∅)
7 dmxp 5927 . . . . . . . 8 ({𝑋} ≠ ∅ → dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴))
82, 6, 7mp2b 10 . . . . . . 7 dom ((𝐵 ∖ dom 𝐴) × {𝑋}) = (𝐵 ∖ dom 𝐴)
98ineq2i 4209 . . . . . 6 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴))
10 disjdif 4471 . . . . . 6 (dom 𝐴 ∩ (𝐵 ∖ dom 𝐴)) = ∅
119, 10eqtri 2761 . . . . 5 (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅
12 funun 6592 . . . . 5 (((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ (dom 𝐴 ∩ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = ∅) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1311, 12mpan2 690 . . . 4 ((Fun 𝐴 ∧ Fun ((𝐵 ∖ dom 𝐴) × {𝑋})) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
141, 5, 13sylancl 587 . . 3 (𝐴 No → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
1514adantr 482 . 2 ((𝐴 No 𝐵 ∈ On) → Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})))
16 dmun 5909 . . . 4 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋}))
178uneq2i 4160 . . . 4 (dom 𝐴 ∪ dom ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
1816, 17eqtri 2761 . . 3 dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴))
19 nodmon 27143 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
20 undif 4481 . . . . . 6 (dom 𝐴𝐵 ↔ (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵)
21 eleq1a 2829 . . . . . . 7 (𝐵 ∈ On → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2221adantl 483 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = 𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
2320, 22biimtrid 241 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴𝐵 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
24 ssdif0 4363 . . . . . 6 (𝐵 ⊆ dom 𝐴 ↔ (𝐵 ∖ dom 𝐴) = ∅)
25 uneq2 4157 . . . . . . . . . 10 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = (dom 𝐴 ∪ ∅))
26 un0 4390 . . . . . . . . . 10 (dom 𝐴 ∪ ∅) = dom 𝐴
2725, 26eqtrdi 2789 . . . . . . . . 9 ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) = dom 𝐴)
2827eleq1d 2819 . . . . . . . 8 ((𝐵 ∖ dom 𝐴) = ∅ → ((dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On ↔ dom 𝐴 ∈ On))
2928biimprcd 249 . . . . . . 7 (dom 𝐴 ∈ On → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
3029adantr 482 . . . . . 6 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ∖ dom 𝐴) = ∅ → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
3124, 30biimtrid 241 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ dom 𝐴 → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On))
32 eloni 6372 . . . . . 6 (dom 𝐴 ∈ On → Ord dom 𝐴)
33 eloni 6372 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
34 ordtri2or2 6461 . . . . . 6 ((Ord dom 𝐴 ∧ Ord 𝐵) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3532, 33, 34syl2an 597 . . . . 5 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴𝐵𝐵 ⊆ dom 𝐴))
3623, 31, 35mpjaod 859 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3719, 36sylan 581 . . 3 ((𝐴 No 𝐵 ∈ On) → (dom 𝐴 ∪ (𝐵 ∖ dom 𝐴)) ∈ On)
3818, 37eqeltrid 2838 . 2 ((𝐴 No 𝐵 ∈ On) → dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On)
39 rnun 6143 . . 3 ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) = (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋}))
40 norn 27144 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
4140adantr 482 . . . 4 ((𝐴 No 𝐵 ∈ On) → ran 𝐴 ⊆ {1o, 2o})
42 rnxpss 6169 . . . . 5 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {𝑋}
43 snssi 4811 . . . . . 6 (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
442, 43ax-mp 5 . . . . 5 {𝑋} ⊆ {1o, 2o}
4542, 44sstri 3991 . . . 4 ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}
46 unss 4184 . . . 4 ((ran 𝐴 ⊆ {1o, 2o} ∧ ran ((𝐵 ∖ dom 𝐴) × {𝑋}) ⊆ {1o, 2o}) ↔ (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
4741, 45, 46sylanblc 590 . . 3 ((𝐴 No 𝐵 ∈ On) → (ran 𝐴 ∪ ran ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
4839, 47eqsstrid 4030 . 2 ((𝐴 No 𝐵 ∈ On) → ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o})
49 elno2 27147 . 2 ((𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No ↔ (Fun (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∧ dom (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ On ∧ ran (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ⊆ {1o, 2o}))
5015, 38, 48, 49syl3anbrc 1344 1 ((𝐴 No 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628  {cpr 4630   × cxp 5674  dom cdm 5676  ran crn 5677  Ord word 6361  Oncon0 6362  Fun wfun 6535   Fn wfn 6536  1oc1o 8456  2oc2o 8457   No csur 27133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  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-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-no 27136
This theorem is referenced by:  noetasuplem1  27226  noetainflem1  27230
  Copyright terms: Public domain W3C validator