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Theorem omeulem2 7609
Description: Lemma for omeu 7611: 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 1087 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐷 ∈ On)
2 eloni 5695 . . . . . 6 (𝐷 ∈ On → Ord 𝐷)
3 ordsucss 6966 . . . . . 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 1085 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐵 ∈ On)
6 suceloni 6961 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
75, 6syl 17 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → suc 𝐵 ∈ On)
8 simp1l 1083 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐴 ∈ On)
9 simp1r 1084 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐴 ≠ ∅)
10 on0eln0 5742 . . . . . . . 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 7596 . . . . . 6 (((suc 𝐵 ∈ On ∧ 𝐷 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝐵𝐷 ↔ (𝐴 ·𝑜 suc 𝐵) ⊆ (𝐴 ·𝑜 𝐷)))
147, 1, 8, 12, 13syl31anc 1326 . . . . 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 7562 . . . . . 6 ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·𝑜 𝐷) ∈ On)
178, 1, 16syl2anc 692 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·𝑜 𝐷) ∈ On)
18 simp3r 1088 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐸𝐴)
19 onelon 5710 . . . . . 6 ((𝐴 ∈ On ∧ 𝐸𝐴) → 𝐸 ∈ On)
208, 18, 19syl2anc 692 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐸 ∈ On)
21 oaword1 7578 . . . . . 6 (((𝐴 ·𝑜 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·𝑜 𝐷) ⊆ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
22 sstr 3596 . . . . . . 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 451 . . . . . 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 692 . . . 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 1086 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐶𝐴)
28 onelon 5710 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶𝐴) → 𝐶 ∈ On)
298, 27, 28syl2anc 692 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐶 ∈ On)
30 omcl 7562 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
318, 5, 30syl2anc 692 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·𝑜 𝐵) ∈ On)
32 oaord 7573 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝐵) ∈ On) → (𝐶𝐴 ↔ ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)))
3332biimpa 501 . . . . 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 1326 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
35 omsuc 7552 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
368, 5, 35syl2anc 692 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
3734, 36eleqtrrd 2707 . . 3 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ (𝐴 ·𝑜 suc 𝐵))
38 ssel 3582 . . 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 477 . . . . 5 ((𝐵 = 𝐷𝐶𝐸) → 𝐶𝐸)
41 oaord 7573 . . . . 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 6613 . . . . . . 7 (𝐵 = 𝐷 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐷))
4443oveq1d 6620 . . . . . 6 (𝐵 = 𝐷 → ((𝐴 ·𝑜 𝐵) +𝑜 𝐸) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
4544adantr 481 . . . . 5 ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐸) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸))
4645eleq2d 2689 . . . 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 1323 . 2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))
4939, 48jaod 395 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wss 3560  c0 3896  Ord word 5684  Oncon0 5685  suc csuc 5687  (class class class)co 6605   +𝑜 coa 7503   ·𝑜 comu 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510  df-omul 7511
This theorem is referenced by:  omopth2  7610
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