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Theorem oaabsb 43284
Description: The right addend absorbs the sum with an ordinal iff that ordinal times omega is less than or equal to the right addend. (Contributed by RP, 19-Feb-2025.)
Assertion
Ref Expression
oaabsb ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵))

Proof of Theorem oaabsb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omelon 9684 . . . . 5 ω ∈ On
2 omcl 8573 . . . . 5 ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ·o ω) ∈ On)
31, 2mpan2 691 . . . 4 (𝐴 ∈ On → (𝐴 ·o ω) ∈ On)
4 oawordex 8594 . . . 4 (((𝐴 ·o ω) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ ∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵))
53, 4sylan 580 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ ∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵))
6 simpl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
76adantr 480 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
83ad2antrr 726 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 ·o ω) ∈ On)
9 simpr 484 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
10 oaass 8598 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐴 ·o ω) ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = (𝐴 +o ((𝐴 ·o ω) +o 𝑥)))
117, 8, 9, 10syl3anc 1370 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = (𝐴 +o ((𝐴 ·o ω) +o 𝑥)))
12 1on 8517 . . . . . . . . . 10 1o ∈ On
13 odi 8616 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 1o ∈ On ∧ ω ∈ On) → (𝐴 ·o (1o +o ω)) = ((𝐴 ·o 1o) +o (𝐴 ·o ω)))
1412, 1, 13mp3an23 1452 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 ·o (1o +o ω)) = ((𝐴 ·o 1o) +o (𝐴 ·o ω)))
15 1oaomeqom 43283 . . . . . . . . . . 11 (1o +o ω) = ω
1615oveq2i 7442 . . . . . . . . . 10 (𝐴 ·o (1o +o ω)) = (𝐴 ·o ω)
1716a1i 11 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 ·o (1o +o ω)) = (𝐴 ·o ω))
18 om1 8579 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
1918oveq1d 7446 . . . . . . . . 9 (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o ω)) = (𝐴 +o (𝐴 ·o ω)))
2014, 17, 193eqtr3rd 2784 . . . . . . . 8 (𝐴 ∈ On → (𝐴 +o (𝐴 ·o ω)) = (𝐴 ·o ω))
2120oveq1d 7446 . . . . . . 7 (𝐴 ∈ On → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = ((𝐴 ·o ω) +o 𝑥))
2221ad2antrr 726 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → ((𝐴 +o (𝐴 ·o ω)) +o 𝑥) = ((𝐴 ·o ω) +o 𝑥))
2311, 22eqtr3d 2777 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = ((𝐴 ·o ω) +o 𝑥))
24 oveq2 7439 . . . . . 6 (((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = (𝐴 +o 𝐵))
25 id 22 . . . . . 6 (((𝐴 ·o ω) +o 𝑥) = 𝐵 → ((𝐴 ·o ω) +o 𝑥) = 𝐵)
2624, 25eqeq12d 2751 . . . . 5 (((𝐴 ·o ω) +o 𝑥) = 𝐵 → ((𝐴 +o ((𝐴 ·o ω) +o 𝑥)) = ((𝐴 ·o ω) +o 𝑥) ↔ (𝐴 +o 𝐵) = 𝐵))
2723, 26syl5ibcom 245 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ On) → (((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
2827rexlimdva 3153 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On ((𝐴 ·o ω) +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
295, 28sylbid 240 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 → (𝐴 +o 𝐵) = 𝐵))
30 limom 7903 . . . . . 6 Lim ω
31 omlim 8570 . . . . . 6 ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
321, 30, 31mpanr12 705 . . . . 5 (𝐴 ∈ On → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
3332ad2antrr 726 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ω) = 𝑥 ∈ ω (𝐴 ·o 𝑥))
34 oveq2 7439 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
3534sseq1d 4027 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o ∅) ⊆ 𝐵))
36 oveq2 7439 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
3736sseq1d 4027 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o 𝑦) ⊆ 𝐵))
38 oveq2 7439 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
3938sseq1d 4027 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ⊆ 𝐵 ↔ (𝐴 ·o suc 𝑦) ⊆ 𝐵))
40 om0 8554 . . . . . . . . . 10 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
41 0ss 4406 . . . . . . . . . 10 ∅ ⊆ 𝐵
4240, 41eqsstrdi 4050 . . . . . . . . 9 (𝐴 ∈ On → (𝐴 ·o ∅) ⊆ 𝐵)
4342ad2antrr 726 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ∅) ⊆ 𝐵)
44 nnon 7893 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ω → 𝑦 ∈ On)
45 omcl 8573 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
466, 44, 45syl2an 596 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ On)
47 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
4847adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → 𝐵 ∈ On)
496adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → 𝐴 ∈ On)
5046, 48, 493jca 1127 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On))
5150expcom 413 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On)))
5251adantrd 491 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On)))
5352imp 406 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → ((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On))
54 oaword 8586 . . . . . . . . . . . 12 (((𝐴 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑦) ⊆ 𝐵 ↔ (𝐴 +o (𝐴 ·o 𝑦)) ⊆ (𝐴 +o 𝐵)))
5553, 54syl 17 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → ((𝐴 ·o 𝑦) ⊆ 𝐵 ↔ (𝐴 +o (𝐴 ·o 𝑦)) ⊆ (𝐴 +o 𝐵)))
5655biimpa 476 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) ⊆ (𝐴 +o 𝐵))
57 simpl 482 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 𝐴 ∈ On)
5812a1i 11 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 1o ∈ On)
5944adantl 481 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
60 odi 8616 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 1o ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (1o +o 𝑦)) = ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)))
6157, 58, 59, 60syl3anc 1370 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o (1o +o 𝑦)) = ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)))
62 1onn 8677 . . . . . . . . . . . . . . . . . . . 20 1o ∈ ω
63 nnacom 8654 . . . . . . . . . . . . . . . . . . . 20 ((1o ∈ ω ∧ 𝑦 ∈ ω) → (1o +o 𝑦) = (𝑦 +o 1o))
6462, 63mpan 690 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ω → (1o +o 𝑦) = (𝑦 +o 1o))
65 oa1suc 8568 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
6644, 65syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ω → (𝑦 +o 1o) = suc 𝑦)
6764, 66eqtrd 2775 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ω → (1o +o 𝑦) = suc 𝑦)
6867oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ω → (𝐴 ·o (1o +o 𝑦)) = (𝐴 ·o suc 𝑦))
6968adantl 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 ·o (1o +o 𝑦)) = (𝐴 ·o suc 𝑦))
7018oveq1d 7446 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)) = (𝐴 +o (𝐴 ·o 𝑦)))
7170adantr 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → ((𝐴 ·o 1o) +o (𝐴 ·o 𝑦)) = (𝐴 +o (𝐴 ·o 𝑦)))
7261, 69, 713eqtr3rd 2784 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
7372expcom 413 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (𝐴 ∈ On → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
7473adantrd 491 . . . . . . . . . . . . 13 (𝑦 ∈ ω → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
7574adantrd 491 . . . . . . . . . . . 12 (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦)))
7675imp 406 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
7776adantr 480 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o (𝐴 ·o 𝑦)) = (𝐴 ·o suc 𝑦))
78 simpr 484 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 +o 𝐵) = 𝐵)
7978adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) → (𝐴 +o 𝐵) = 𝐵)
8079adantr 480 . . . . . . . . . 10 (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)
8156, 77, 803sstr3d 4042 . . . . . . . . 9 (((𝑦 ∈ ω ∧ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵)) ∧ (𝐴 ·o 𝑦) ⊆ 𝐵) → (𝐴 ·o suc 𝑦) ⊆ 𝐵)
8281exp31 419 . . . . . . . 8 (𝑦 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ((𝐴 ·o 𝑦) ⊆ 𝐵 → (𝐴 ·o suc 𝑦) ⊆ 𝐵)))
8335, 37, 39, 43, 82finds2 7921 . . . . . . 7 (𝑥 ∈ ω → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o 𝑥) ⊆ 𝐵))
8483com12 32 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝑥 ∈ ω → (𝐴 ·o 𝑥) ⊆ 𝐵))
8584ralrimiv 3143 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
86 iunss 5050 . . . . 5 ( 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵 ↔ ∀𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
8785, 86sylibr 234 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → 𝑥 ∈ ω (𝐴 ·o 𝑥) ⊆ 𝐵)
8833, 87eqsstrd 4034 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 +o 𝐵) = 𝐵) → (𝐴 ·o ω) ⊆ 𝐵)
8988ex 412 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = 𝐵 → (𝐴 ·o ω) ⊆ 𝐵))
9029, 89impbid 212 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o ω) ⊆ 𝐵 ↔ (𝐴 +o 𝐵) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  c0 4339   ciun 4996  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431  ωcom 7887  1oc1o 8498   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by: (None)
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