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Theorem omeulem2 7708
Description: Lemma for omeu 7710: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
omeulem2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))

Proof of Theorem omeulem2
StepHypRef Expression
1 simp3l 1109 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐷 ∈ On)
2 eloni 5771 . . . . . 6 (𝐷 ∈ On → Ord 𝐷)
3 ordsucss 7060 . . . . . 6 (Ord 𝐷 → (𝐵𝐷 → suc 𝐵𝐷))
41, 2, 33syl 18 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐵𝐷 → suc 𝐵𝐷))
5 simp2l 1107 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐵 ∈ On)
6 suceloni 7055 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
75, 6syl 17 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → suc 𝐵 ∈ On)
8 simp1l 1105 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐴 ∈ On)
9 simp1r 1106 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐴 ≠ ∅)
10 on0eln0 5818 . . . . . . . 8 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
118, 10syl 17 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (∅ ∈ 𝐴𝐴 ≠ ∅))
129, 11mpbird 247 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ∅ ∈ 𝐴)
13 omword 7695 . . . . . 6 (((suc 𝐵 ∈ On ∧ 𝐷 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝐵𝐷 ↔ (𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷)))
147, 1, 8, 12, 13syl31anc 1369 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (suc 𝐵𝐷 ↔ (𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷)))
154, 14sylibd 229 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐵𝐷 → (𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷)))
16 omcl 7661 . . . . . 6 ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·𝑜 𝐷) ∈ On)
178, 1, 16syl2anc 694 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·𝑜 𝐷) ∈ On)
18 simp3r 1110 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐸𝐴)
19 onelon 5786 . . . . . 6 ((𝐴 ∈ On ∧ 𝐸𝐴) → 𝐸 ∈ On)
208, 18, 19syl2anc 694 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐸 ∈ On)
21 oaword1 7677 . . . . . 6 (((𝐴 ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·𝑜 𝐷) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
22 sstr 3644 . . . . . . 7 (((𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷) ∧ (𝐴 ·𝑜 𝐷) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)) → (𝐴 ·𝑜 suc 𝐵) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
2322expcom 450 . . . . . 6 ((𝐴 ·𝑜 𝐷) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) → ((𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷) → (𝐴 ·𝑜 suc 𝐵) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2421, 23syl 17 . . . . 5 (((𝐴 ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷) → (𝐴 ·𝑜 suc 𝐵) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2517, 20, 24syl2anc 694 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷) → (𝐴 ·𝑜 suc 𝐵) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
2615, 25syld 47 . . 3 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐵𝐷 → (𝐴 ·𝑜 suc 𝐵) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
27 simp2r 1108 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐶𝐴)
28 onelon 5786 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶𝐴) → 𝐶 ∈ On)
298, 27, 28syl2anc 694 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐶 ∈ On)
30 omcl 7661 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
318, 5, 30syl2anc 694 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·𝑜 𝐵) ∈ On)
32 oaord 7672 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝐵) ∈ On) → (𝐶𝐴 ↔ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
3332biimpa 500 . . . . 5 (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝐵) ∈ On) ∧ 𝐶𝐴) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
3429, 8, 31, 27, 33syl31anc 1369 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
35 omsuc 7651 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
368, 5, 35syl2anc 694 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
3734, 36eleqtrrd 2733 . . 3 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ (𝐴 ·𝑜 suc 𝐵))
38 ssel 3630 . . 3 ((𝐴 ·𝑜 suc 𝐵) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) → (((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ (𝐴 ·𝑜 suc 𝐵) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
3926, 37, 38syl6ci 71 . 2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐵𝐷 → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
40 simpr 476 . . . . 5 ((𝐵 = 𝐷𝐶𝐸) → 𝐶𝐸)
41 oaord 7672 . . . . 5 ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·𝑜 𝐵) ∈ On) → (𝐶𝐸 ↔ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐸)))
4240, 41syl5ib 234 . . . 4 ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·𝑜 𝐵) ∈ On) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐸)))
43 oveq2 6698 . . . . . . 7 (𝐵 = 𝐷 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐷))
4443oveq1d 6705 . . . . . 6 (𝐵 = 𝐷 → ((𝐴 ·𝑜 𝐵) +𝑜 𝐸) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
4544adantr 480 . . . . 5 ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐸) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
4645eleq2d 2716 . . . 4 ((𝐵 = 𝐷𝐶𝐸) → (((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐸) ↔ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
4742, 46mpbidi 231 . . 3 ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·𝑜 𝐵) ∈ On) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
4829, 20, 31, 47syl3anc 1366 . 2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
4939, 48jaod 394 1 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wss 3607  c0 3948  Ord word 5760  Oncon0 5761  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610
This theorem is referenced by:  omopth2  7709
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