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Theorem omeulem2 8621
Description: Lemma for omeu 8623: 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 1202 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐷 ∈ On)
2 eloni 6394 . . . . . 6 (𝐷 ∈ On → Ord 𝐷)
3 ordsucss 7838 . . . . . 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 1200 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐵 ∈ On)
6 onsuc 7831 . . . . . . 7 (𝐵 ∈ On → suc 𝐵 ∈ On)
75, 6syl 17 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → suc 𝐵 ∈ On)
8 simp1l 1198 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐴 ∈ On)
9 simp1r 1199 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐴 ≠ ∅)
10 on0eln0 6440 . . . . . . . 8 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
118, 10syl 17 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (∅ ∈ 𝐴𝐴 ≠ ∅))
129, 11mpbird 257 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ∅ ∈ 𝐴)
13 omword 8608 . . . . . 6 (((suc 𝐵 ∈ On ∧ 𝐷 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝐵𝐷 ↔ (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
147, 1, 8, 12, 13syl31anc 1375 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (suc 𝐵𝐷 ↔ (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
154, 14sylibd 239 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐵𝐷 → (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
16 omcl 8574 . . . . . 6 ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
178, 1, 16syl2anc 584 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·o 𝐷) ∈ On)
18 simp3r 1203 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐸𝐴)
19 onelon 6409 . . . . . 6 ((𝐴 ∈ On ∧ 𝐸𝐴) → 𝐸 ∈ On)
208, 18, 19syl2anc 584 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐸 ∈ On)
21 oaword1 8590 . . . . . 6 (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
22 sstr 3992 . . . . . . 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 413 . . . . . 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 584 . . . 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 1201 . . . . . 6 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐶𝐴)
28 onelon 6409 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶𝐴) → 𝐶 ∈ On)
298, 27, 28syl2anc 584 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → 𝐶 ∈ On)
30 omcl 8574 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
318, 5, 30syl2anc 584 . . . . 5 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·o 𝐵) ∈ On)
32 oaord 8585 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶𝐴 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴)))
3332biimpa 476 . . . . 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 1375 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
35 omsuc 8564 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
368, 5, 35syl2anc 584 . . . 4 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
3734, 36eleqtrrd 2844 . . 3 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵))
38 ssel 3977 . . 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 484 . . . . 5 ((𝐵 = 𝐷𝐶𝐸) → 𝐶𝐸)
41 oaord 8585 . . . . 5 ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶𝐸 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
4240, 41imbitrid 244 . . . 4 ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
43 oveq2 7439 . . . . . . 7 (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
4443oveq1d 7446 . . . . . 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 2827 . . . 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 241 . . 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 1373 . 2 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵 = 𝐷𝐶𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
4939, 48jaod 860 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wss 3951  c0 4333  Ord word 6383  Oncon0 6384  suc csuc 6386  (class class class)co 7431   +o coa 8503   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511
This theorem is referenced by:  omopth2  8622
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