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Theorem oeeui 8220
 Description: The division algorithm for ordinal exponentiation. (This version of oeeu 8221 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8203.) (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
oeeu.1 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}
oeeu.2 𝑃 = (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵))
oeeu.3 𝑌 = (1st𝑃)
oeeu.4 𝑍 = (2nd𝑃)
Assertion
Ref Expression
oeeui ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o) ∧ 𝐸 ∈ (𝐴o 𝐶)) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍)))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝑃(𝑥,𝑦,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)   𝑋(𝑥)   𝑌(𝑥,𝑦,𝑧,𝑤)   𝑍(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem oeeui
Dummy variables 𝑎 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 4101 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
21adantr 483 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
32ad2antrr 724 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐴 ∈ On)
4 simprl 769 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐶 ∈ On)
5 oecl 8154 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
63, 4, 5syl2anc 586 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝐶) ∈ On)
7 om1 8160 . . . . . . . . . . . . . . 15 ((𝐴o 𝐶) ∈ On → ((𝐴o 𝐶) ·o 1o) = (𝐴o 𝐶))
86, 7syl 17 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o 1o) = (𝐴o 𝐶))
9 df1o2 8108 . . . . . . . . . . . . . . . 16 1o = {∅}
10 dif1o 8117 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (𝐴 ∖ 1o) ↔ (𝐷𝐴𝐷 ≠ ∅))
1110simprbi 499 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ (𝐴 ∖ 1o) → 𝐷 ≠ ∅)
1211ad2antll 727 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐷 ≠ ∅)
13 eldifi 4101 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (𝐴 ∖ 1o) → 𝐷𝐴)
1413ad2antll 727 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐷𝐴)
15 onelon 6209 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝐷𝐴) → 𝐷 ∈ On)
163, 14, 15syl2anc 586 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐷 ∈ On)
17 on0eln0 6239 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ On → (∅ ∈ 𝐷𝐷 ≠ ∅))
1816, 17syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (∅ ∈ 𝐷𝐷 ≠ ∅))
1912, 18mpbird 259 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ∅ ∈ 𝐷)
2019snssd 4734 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → {∅} ⊆ 𝐷)
219, 20eqsstrid 4013 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 1o𝐷)
22 1on 8101 . . . . . . . . . . . . . . . . 17 1o ∈ On
2322a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 1o ∈ On)
24 omwordi 8189 . . . . . . . . . . . . . . . 16 ((1o ∈ On ∧ 𝐷 ∈ On ∧ (𝐴o 𝐶) ∈ On) → (1o𝐷 → ((𝐴o 𝐶) ·o 1o) ⊆ ((𝐴o 𝐶) ·o 𝐷)))
2523, 16, 6, 24syl3anc 1366 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (1o𝐷 → ((𝐴o 𝐶) ·o 1o) ⊆ ((𝐴o 𝐶) ·o 𝐷)))
2621, 25mpd 15 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o 1o) ⊆ ((𝐴o 𝐶) ·o 𝐷))
278, 26eqsstrrd 4004 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝐶) ⊆ ((𝐴o 𝐶) ·o 𝐷))
28 omcl 8153 . . . . . . . . . . . . . . . 16 (((𝐴o 𝐶) ∈ On ∧ 𝐷 ∈ On) → ((𝐴o 𝐶) ·o 𝐷) ∈ On)
296, 16, 28syl2anc 586 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o 𝐷) ∈ On)
30 simplrl 775 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐸 ∈ (𝐴o 𝐶))
31 onelon 6209 . . . . . . . . . . . . . . . 16 (((𝐴o 𝐶) ∈ On ∧ 𝐸 ∈ (𝐴o 𝐶)) → 𝐸 ∈ On)
326, 30, 31syl2anc 586 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐸 ∈ On)
33 oaword1 8170 . . . . . . . . . . . . . . 15 ((((𝐴o 𝐶) ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴o 𝐶) ·o 𝐷) ⊆ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸))
3429, 32, 33syl2anc 586 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o 𝐷) ⊆ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸))
35 simplrr 776 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)
3634, 35sseqtrd 4005 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o 𝐷) ⊆ 𝐵)
3727, 36sstrd 3975 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝐶) ⊆ 𝐵)
38 oeeu.1 . . . . . . . . . . . . . . 15 𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴o 𝑥)}
3938oeeulem 8219 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝑋 ∈ On ∧ (𝐴o 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴o suc 𝑋)))
4039simp3d 1139 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴o suc 𝑋))
4140ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐵 ∈ (𝐴o suc 𝑋))
4239simp1d 1137 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
4342ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝑋 ∈ On)
44 suceloni 7520 . . . . . . . . . . . . . . 15 (𝑋 ∈ On → suc 𝑋 ∈ On)
4543, 44syl 17 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → suc 𝑋 ∈ On)
46 oecl 8154 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ suc 𝑋 ∈ On) → (𝐴o suc 𝑋) ∈ On)
473, 45, 46syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o suc 𝑋) ∈ On)
48 ontr2 6231 . . . . . . . . . . . . 13 (((𝐴o 𝐶) ∈ On ∧ (𝐴o suc 𝑋) ∈ On) → (((𝐴o 𝐶) ⊆ 𝐵𝐵 ∈ (𝐴o suc 𝑋)) → (𝐴o 𝐶) ∈ (𝐴o suc 𝑋)))
496, 47, 48syl2anc 586 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (((𝐴o 𝐶) ⊆ 𝐵𝐵 ∈ (𝐴o suc 𝑋)) → (𝐴o 𝐶) ∈ (𝐴o suc 𝑋)))
5037, 41, 49mp2and 697 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝐶) ∈ (𝐴o suc 𝑋))
51 simplll 773 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐴 ∈ (On ∖ 2o))
52 oeord 8206 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ suc 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐶 ∈ suc 𝑋 ↔ (𝐴o 𝐶) ∈ (𝐴o suc 𝑋)))
534, 45, 51, 52syl3anc 1366 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐶 ∈ suc 𝑋 ↔ (𝐴o 𝐶) ∈ (𝐴o suc 𝑋)))
5450, 53mpbird 259 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐶 ∈ suc 𝑋)
55 onsssuc 6271 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ 𝑋 ∈ On) → (𝐶𝑋𝐶 ∈ suc 𝑋))
564, 43, 55syl2anc 586 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐶𝑋𝐶 ∈ suc 𝑋))
5754, 56mpbird 259 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐶𝑋)
5839simp2d 1138 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o 𝑋) ⊆ 𝐵)
5958ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝑋) ⊆ 𝐵)
60 eloni 6194 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → Ord 𝐴)
613, 60syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → Ord 𝐴)
62 ordsucss 7525 . . . . . . . . . . . . . . . 16 (Ord 𝐴 → (𝐷𝐴 → suc 𝐷𝐴))
6361, 14, 62sylc 65 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → suc 𝐷𝐴)
64 suceloni 7520 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ On → suc 𝐷 ∈ On)
6516, 64syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → suc 𝐷 ∈ On)
66 dif20el 8122 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
6751, 66syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ∅ ∈ 𝐴)
68 oen0 8204 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐶))
693, 4, 67, 68syl21anc 835 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ∅ ∈ (𝐴o 𝐶))
70 omword 8188 . . . . . . . . . . . . . . . 16 (((suc 𝐷 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴o 𝐶) ∈ On) ∧ ∅ ∈ (𝐴o 𝐶)) → (suc 𝐷𝐴 ↔ ((𝐴o 𝐶) ·o suc 𝐷) ⊆ ((𝐴o 𝐶) ·o 𝐴)))
7165, 3, 6, 69, 70syl31anc 1368 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (suc 𝐷𝐴 ↔ ((𝐴o 𝐶) ·o suc 𝐷) ⊆ ((𝐴o 𝐶) ·o 𝐴)))
7263, 71mpbid 234 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o suc 𝐷) ⊆ ((𝐴o 𝐶) ·o 𝐴))
73 oaord 8165 . . . . . . . . . . . . . . . . . 18 ((𝐸 ∈ On ∧ (𝐴o 𝐶) ∈ On ∧ ((𝐴o 𝐶) ·o 𝐷) ∈ On) → (𝐸 ∈ (𝐴o 𝐶) ↔ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) ∈ (((𝐴o 𝐶) ·o 𝐷) +o (𝐴o 𝐶))))
7432, 6, 29, 73syl3anc 1366 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐸 ∈ (𝐴o 𝐶) ↔ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) ∈ (((𝐴o 𝐶) ·o 𝐷) +o (𝐴o 𝐶))))
7530, 74mpbid 234 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) ∈ (((𝐴o 𝐶) ·o 𝐷) +o (𝐴o 𝐶)))
7635, 75eqeltrrd 2912 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐵 ∈ (((𝐴o 𝐶) ·o 𝐷) +o (𝐴o 𝐶)))
77 odi 8197 . . . . . . . . . . . . . . . . 17 (((𝐴o 𝐶) ∈ On ∧ 𝐷 ∈ On ∧ 1o ∈ On) → ((𝐴o 𝐶) ·o (𝐷 +o 1o)) = (((𝐴o 𝐶) ·o 𝐷) +o ((𝐴o 𝐶) ·o 1o)))
786, 16, 23, 77syl3anc 1366 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o (𝐷 +o 1o)) = (((𝐴o 𝐶) ·o 𝐷) +o ((𝐴o 𝐶) ·o 1o)))
79 oa1suc 8148 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ On → (𝐷 +o 1o) = suc 𝐷)
8016, 79syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐷 +o 1o) = suc 𝐷)
8180oveq2d 7164 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o (𝐷 +o 1o)) = ((𝐴o 𝐶) ·o suc 𝐷))
828oveq2d 7164 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (((𝐴o 𝐶) ·o 𝐷) +o ((𝐴o 𝐶) ·o 1o)) = (((𝐴o 𝐶) ·o 𝐷) +o (𝐴o 𝐶)))
8378, 81, 823eqtr3d 2862 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → ((𝐴o 𝐶) ·o suc 𝐷) = (((𝐴o 𝐶) ·o 𝐷) +o (𝐴o 𝐶)))
8476, 83eleqtrrd 2914 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐵 ∈ ((𝐴o 𝐶) ·o suc 𝐷))
8572, 84sseldd 3966 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐵 ∈ ((𝐴o 𝐶) ·o 𝐴))
86 oesuc 8144 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
873, 4, 86syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
8885, 87eleqtrrd 2914 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐵 ∈ (𝐴o suc 𝐶))
89 oecl 8154 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
903, 43, 89syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝑋) ∈ On)
91 suceloni 7520 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → suc 𝐶 ∈ On)
9291ad2antrl 726 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → suc 𝐶 ∈ On)
93 oecl 8154 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴o suc 𝐶) ∈ On)
943, 92, 93syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o suc 𝐶) ∈ On)
95 ontr2 6231 . . . . . . . . . . . . 13 (((𝐴o 𝑋) ∈ On ∧ (𝐴o suc 𝐶) ∈ On) → (((𝐴o 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴o suc 𝐶)) → (𝐴o 𝑋) ∈ (𝐴o suc 𝐶)))
9690, 94, 95syl2anc 586 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (((𝐴o 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴o suc 𝐶)) → (𝐴o 𝑋) ∈ (𝐴o suc 𝐶)))
9759, 88, 96mp2and 697 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐴o 𝑋) ∈ (𝐴o suc 𝐶))
98 oeord 8206 . . . . . . . . . . . 12 ((𝑋 ∈ On ∧ suc 𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑋 ∈ suc 𝐶 ↔ (𝐴o 𝑋) ∈ (𝐴o suc 𝐶)))
9943, 92, 51, 98syl3anc 1366 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝑋 ∈ suc 𝐶 ↔ (𝐴o 𝑋) ∈ (𝐴o suc 𝐶)))
10097, 99mpbird 259 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝑋 ∈ suc 𝐶)
101 onsssuc 6271 . . . . . . . . . . 11 ((𝑋 ∈ On ∧ 𝐶 ∈ On) → (𝑋𝐶𝑋 ∈ suc 𝐶))
10243, 4, 101syl2anc 586 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝑋𝐶𝑋 ∈ suc 𝐶))
103100, 102mpbird 259 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝑋𝐶)
10457, 103eqssd 3982 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → 𝐶 = 𝑋)
105104, 16jca 514 . . . . . . 7 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o))) → (𝐶 = 𝑋𝐷 ∈ On))
106 simprl 769 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐶 = 𝑋)
10742ad2antrr 724 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝑋 ∈ On)
108106, 107eqeltrd 2911 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐶 ∈ On)
1092ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐴 ∈ On)
110109, 108, 5syl2anc 586 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o 𝐶) ∈ On)
111 simprr 771 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷 ∈ On)
112110, 111, 28syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ·o 𝐷) ∈ On)
113 simplrl 775 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐸 ∈ (𝐴o 𝐶))
114110, 113, 31syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐸 ∈ On)
115112, 114, 33syl2anc 586 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ·o 𝐷) ⊆ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸))
116 simplrr 776 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)
117115, 116sseqtrd 4005 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ·o 𝐷) ⊆ 𝐵)
11840ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐵 ∈ (𝐴o suc 𝑋))
119 suceq 6249 . . . . . . . . . . . . . . 15 (𝐶 = 𝑋 → suc 𝐶 = suc 𝑋)
120119ad2antrl 726 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → suc 𝐶 = suc 𝑋)
121120oveq2d 7164 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o suc 𝐶) = (𝐴o suc 𝑋))
122109, 108, 86syl2anc 586 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
123121, 122eqtr3d 2856 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o suc 𝑋) = ((𝐴o 𝐶) ·o 𝐴))
124118, 123eleqtrd 2913 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐵 ∈ ((𝐴o 𝐶) ·o 𝐴))
125 omcl 8153 . . . . . . . . . . . . 13 (((𝐴o 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴o 𝐶) ·o 𝐴) ∈ On)
126110, 109, 125syl2anc 586 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ·o 𝐴) ∈ On)
127 ontr2 6231 . . . . . . . . . . . 12 ((((𝐴o 𝐶) ·o 𝐷) ∈ On ∧ ((𝐴o 𝐶) ·o 𝐴) ∈ On) → ((((𝐴o 𝐶) ·o 𝐷) ⊆ 𝐵𝐵 ∈ ((𝐴o 𝐶) ·o 𝐴)) → ((𝐴o 𝐶) ·o 𝐷) ∈ ((𝐴o 𝐶) ·o 𝐴)))
128112, 126, 127syl2anc 586 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((((𝐴o 𝐶) ·o 𝐷) ⊆ 𝐵𝐵 ∈ ((𝐴o 𝐶) ·o 𝐴)) → ((𝐴o 𝐶) ·o 𝐷) ∈ ((𝐴o 𝐶) ·o 𝐴)))
129117, 124, 128mp2and 697 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ·o 𝐷) ∈ ((𝐴o 𝐶) ·o 𝐴))
13066adantr 483 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∅ ∈ 𝐴)
131130ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ∅ ∈ 𝐴)
132109, 108, 131, 68syl21anc 835 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ∅ ∈ (𝐴o 𝐶))
133 omord2 8185 . . . . . . . . . . 11 (((𝐷 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴o 𝐶) ∈ On) ∧ ∅ ∈ (𝐴o 𝐶)) → (𝐷𝐴 ↔ ((𝐴o 𝐶) ·o 𝐷) ∈ ((𝐴o 𝐶) ·o 𝐴)))
134111, 109, 110, 132, 133syl31anc 1368 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐷𝐴 ↔ ((𝐴o 𝐶) ·o 𝐷) ∈ ((𝐴o 𝐶) ·o 𝐴)))
135129, 134mpbird 259 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷𝐴)
136106oveq2d 7164 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o 𝐶) = (𝐴o 𝑋))
13758ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o 𝑋) ⊆ 𝐵)
138136, 137eqsstrd 4003 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐴o 𝐶) ⊆ 𝐵)
139 eldifi 4101 . . . . . . . . . . . . . 14 (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
140139adantl 484 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
141140ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐵 ∈ On)
142 ontri1 6218 . . . . . . . . . . . 12 (((𝐴o 𝐶) ∈ On ∧ 𝐵 ∈ On) → ((𝐴o 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴o 𝐶)))
143110, 141, 142syl2anc 586 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴o 𝐶)))
144138, 143mpbid 234 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ¬ 𝐵 ∈ (𝐴o 𝐶))
145 om0 8134 . . . . . . . . . . . . . . . . 17 ((𝐴o 𝐶) ∈ On → ((𝐴o 𝐶) ·o ∅) = ∅)
146110, 145syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((𝐴o 𝐶) ·o ∅) = ∅)
147146oveq1d 7163 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴o 𝐶) ·o ∅) +o 𝐸) = (∅ +o 𝐸))
148 oa0r 8155 . . . . . . . . . . . . . . . 16 (𝐸 ∈ On → (∅ +o 𝐸) = 𝐸)
149114, 148syl 17 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (∅ +o 𝐸) = 𝐸)
150147, 149eqtrd 2854 . . . . . . . . . . . . . 14 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴o 𝐶) ·o ∅) +o 𝐸) = 𝐸)
151150, 113eqeltrd 2911 . . . . . . . . . . . . 13 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (((𝐴o 𝐶) ·o ∅) +o 𝐸) ∈ (𝐴o 𝐶))
152 oveq2 7156 . . . . . . . . . . . . . . 15 (𝐷 = ∅ → ((𝐴o 𝐶) ·o 𝐷) = ((𝐴o 𝐶) ·o ∅))
153152oveq1d 7163 . . . . . . . . . . . . . 14 (𝐷 = ∅ → (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = (((𝐴o 𝐶) ·o ∅) +o 𝐸))
154153eleq1d 2895 . . . . . . . . . . . . 13 (𝐷 = ∅ → ((((𝐴o 𝐶) ·o 𝐷) +o 𝐸) ∈ (𝐴o 𝐶) ↔ (((𝐴o 𝐶) ·o ∅) +o 𝐸) ∈ (𝐴o 𝐶)))
155151, 154syl5ibrcom 249 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐷 = ∅ → (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) ∈ (𝐴o 𝐶)))
156116eleq1d 2895 . . . . . . . . . . . 12 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → ((((𝐴o 𝐶) ·o 𝐷) +o 𝐸) ∈ (𝐴o 𝐶) ↔ 𝐵 ∈ (𝐴o 𝐶)))
157155, 156sylibd 241 . . . . . . . . . . 11 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐷 = ∅ → 𝐵 ∈ (𝐴o 𝐶)))
158157necon3bd 3028 . . . . . . . . . 10 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (¬ 𝐵 ∈ (𝐴o 𝐶) → 𝐷 ≠ ∅))
159144, 158mpd 15 . . . . . . . . 9 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷 ≠ ∅)
160135, 159, 10sylanbrc 585 . . . . . . . 8 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → 𝐷 ∈ (𝐴 ∖ 1o))
161108, 160jca 514 . . . . . . 7 ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ∧ (𝐶 = 𝑋𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)))
162105, 161impbida 799 . . . . . 6 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)) ↔ (𝐶 = 𝑋𝐷 ∈ On)))
163162ex 415 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) → ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)) ↔ (𝐶 = 𝑋𝐷 ∈ On))))
164163pm5.32rd 580 . . . 4 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ↔ ((𝐶 = 𝑋𝐷 ∈ On) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵))))
165 anass 471 . . . 4 (((𝐶 = 𝑋𝐷 ∈ On) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵))))
166164, 165syl6bb 289 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)))))
167 3anass 1090 . . . . . 6 ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)))
168 oveq2 7156 . . . . . . . 8 (𝐶 = 𝑋 → (𝐴o 𝐶) = (𝐴o 𝑋))
169168eleq2d 2896 . . . . . . 7 (𝐶 = 𝑋 → (𝐸 ∈ (𝐴o 𝐶) ↔ 𝐸 ∈ (𝐴o 𝑋)))
170168oveq1d 7163 . . . . . . . . 9 (𝐶 = 𝑋 → ((𝐴o 𝐶) ·o 𝐷) = ((𝐴o 𝑋) ·o 𝐷))
171170oveq1d 7163 . . . . . . . 8 (𝐶 = 𝑋 → (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = (((𝐴o 𝑋) ·o 𝐷) +o 𝐸))
172171eqeq1d 2821 . . . . . . 7 (𝐶 = 𝑋 → ((((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵 ↔ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵))
173169, 1723anbi23d 1433 . . . . . 6 (𝐶 = 𝑋 → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝑋) ∧ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵)))
174167, 173syl5bbr 287 . . . . 5 (𝐶 = 𝑋 → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ↔ (𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝑋) ∧ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵)))
1752, 42, 89syl2anc 586 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o 𝑋) ∈ On)
176 oen0 8204 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑋 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝑋))
1772, 42, 130, 176syl21anc 835 . . . . . . 7 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∅ ∈ (𝐴o 𝑋))
178177ne0d 4299 . . . . . 6 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴o 𝑋) ≠ ∅)
179 omeu 8203 . . . . . . 7 (((𝐴o 𝑋) ∈ On ∧ 𝐵 ∈ On ∧ (𝐴o 𝑋) ≠ ∅) → ∃!𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
180 oeeu.2 . . . . . . . . 9 𝑃 = (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵))
181 opeq1 4795 . . . . . . . . . . . . . 14 (𝑦 = 𝑑 → ⟨𝑦, 𝑧⟩ = ⟨𝑑, 𝑧⟩)
182181eqeq2d 2830 . . . . . . . . . . . . 13 (𝑦 = 𝑑 → (𝑤 = ⟨𝑦, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑧⟩))
183 oveq2 7156 . . . . . . . . . . . . . . 15 (𝑦 = 𝑑 → ((𝐴o 𝑋) ·o 𝑦) = ((𝐴o 𝑋) ·o 𝑑))
184183oveq1d 7163 . . . . . . . . . . . . . 14 (𝑦 = 𝑑 → (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = (((𝐴o 𝑋) ·o 𝑑) +o 𝑧))
185184eqeq1d 2821 . . . . . . . . . . . . 13 (𝑦 = 𝑑 → ((((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵 ↔ (((𝐴o 𝑋) ·o 𝑑) +o 𝑧) = 𝐵))
186182, 185anbi12d 632 . . . . . . . . . . . 12 (𝑦 = 𝑑 → ((𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵) ↔ (𝑤 = ⟨𝑑, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑧) = 𝐵)))
187 opeq2 4796 . . . . . . . . . . . . . 14 (𝑧 = 𝑒 → ⟨𝑑, 𝑧⟩ = ⟨𝑑, 𝑒⟩)
188187eqeq2d 2830 . . . . . . . . . . . . 13 (𝑧 = 𝑒 → (𝑤 = ⟨𝑑, 𝑧⟩ ↔ 𝑤 = ⟨𝑑, 𝑒⟩))
189 oveq2 7156 . . . . . . . . . . . . . 14 (𝑧 = 𝑒 → (((𝐴o 𝑋) ·o 𝑑) +o 𝑧) = (((𝐴o 𝑋) ·o 𝑑) +o 𝑒))
190189eqeq1d 2821 . . . . . . . . . . . . 13 (𝑧 = 𝑒 → ((((𝐴o 𝑋) ·o 𝑑) +o 𝑧) = 𝐵 ↔ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
191188, 190anbi12d 632 . . . . . . . . . . . 12 (𝑧 = 𝑒 → ((𝑤 = ⟨𝑑, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑧) = 𝐵) ↔ (𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵)))
192186, 191cbvrex2vw 3461 . . . . . . . . . . 11 (∃𝑦 ∈ On ∃𝑧 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
193 eqeq1 2823 . . . . . . . . . . . . 13 (𝑤 = 𝑎 → (𝑤 = ⟨𝑑, 𝑒⟩ ↔ 𝑎 = ⟨𝑑, 𝑒⟩))
194193anbi1d 631 . . . . . . . . . . . 12 (𝑤 = 𝑎 → ((𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵) ↔ (𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵)))
1951942rexbidv 3298 . . . . . . . . . . 11 (𝑤 = 𝑎 → (∃𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵)))
196192, 195syl5bb 285 . . . . . . . . . 10 (𝑤 = 𝑎 → (∃𝑦 ∈ On ∃𝑧 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵) ↔ ∃𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵)))
197196cbviotavw 6315 . . . . . . . . 9 (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴o 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵)) = (℩𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
198180, 197eqtri 2842 . . . . . . . 8 𝑃 = (℩𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵))
199 oeeu.3 . . . . . . . 8 𝑌 = (1st𝑃)
200 oeeu.4 . . . . . . . 8 𝑍 = (2nd𝑃)
201 oveq2 7156 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝐴o 𝑋) ·o 𝑑) = ((𝐴o 𝑋) ·o 𝐷))
202201oveq1d 7163 . . . . . . . . 9 (𝑑 = 𝐷 → (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = (((𝐴o 𝑋) ·o 𝐷) +o 𝑒))
203202eqeq1d 2821 . . . . . . . 8 (𝑑 = 𝐷 → ((((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵 ↔ (((𝐴o 𝑋) ·o 𝐷) +o 𝑒) = 𝐵))
204 oveq2 7156 . . . . . . . . 9 (𝑒 = 𝐸 → (((𝐴o 𝑋) ·o 𝐷) +o 𝑒) = (((𝐴o 𝑋) ·o 𝐷) +o 𝐸))
205204eqeq1d 2821 . . . . . . . 8 (𝑒 = 𝐸 → ((((𝐴o 𝑋) ·o 𝐷) +o 𝑒) = 𝐵 ↔ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵))
206198, 199, 200, 203, 205opiota 7749 . . . . . . 7 (∃!𝑎𝑑 ∈ On ∃𝑒 ∈ (𝐴o 𝑋)(𝑎 = ⟨𝑑, 𝑒⟩ ∧ (((𝐴o 𝑋) ·o 𝑑) +o 𝑒) = 𝐵) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝑋) ∧ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
207179, 206syl 17 . . . . . 6 (((𝐴o 𝑋) ∈ On ∧ 𝐵 ∈ On ∧ (𝐴o 𝑋) ≠ ∅) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝑋) ∧ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
208175, 140, 178, 207syl3anc 1366 . . . . 5 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐷 ∈ On ∧ 𝐸 ∈ (𝐴o 𝑋) ∧ (((𝐴o 𝑋) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
209174, 208sylan9bbr 513 . . . 4 (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ 𝐶 = 𝑋) → ((𝐷 ∈ On ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ↔ (𝐷 = 𝑌𝐸 = 𝑍)))
210209pm5.32da 581 . . 3 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐶 = 𝑋 ∧ (𝐷 ∈ On ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵))) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌𝐸 = 𝑍))))
211166, 210bitrd 281 . 2 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌𝐸 = 𝑍))))
212 3an4anass 1100 . 2 (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o) ∧ 𝐸 ∈ (𝐴o 𝐶)) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ ((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o)) ∧ (𝐸 ∈ (𝐴o 𝐶) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵)))
213 3anass 1090 . 2 ((𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍) ↔ (𝐶 = 𝑋 ∧ (𝐷 = 𝑌𝐸 = 𝑍)))
214211, 212, 2133bitr4g 316 1 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o) ∧ 𝐸 ∈ (𝐴o 𝐶)) ∧ (((𝐴o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  ∃!weu 2647   ≠ wne 3014  ∃wrex 3137  {crab 3140   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  {csn 4559  ⟨cop 4565  ∪ cuni 4830  ∩ cint 4867  Ord word 6183  Oncon0 6184  suc csuc 6186  ℩cio 6305  ‘cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  1oc1o 8087  2oc2o 8088   +o coa 8091   ·o comu 8092   ↑o coe 8093 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-omul 8099  df-oexp 8100 This theorem is referenced by:  oeeu  8221  cantnflem3  9146  cantnflem4  9147
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