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Theorem oewordri 8613
Description: Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
Assertion
Ref Expression
oewordri ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))

Proof of Theorem oewordri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7427 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
2 oveq2 7427 . . . . 5 (𝑥 = ∅ → (𝐵o 𝑥) = (𝐵o ∅))
31, 2sseq12d 4010 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o ∅) ⊆ (𝐵o ∅)))
4 oveq2 7427 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
5 oveq2 7427 . . . . 5 (𝑥 = 𝑦 → (𝐵o 𝑥) = (𝐵o 𝑦))
64, 5sseq12d 4010 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)))
7 oveq2 7427 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
8 oveq2 7427 . . . . 5 (𝑥 = suc 𝑦 → (𝐵o 𝑥) = (𝐵o suc 𝑦))
97, 8sseq12d 4010 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦)))
10 oveq2 7427 . . . . 5 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
11 oveq2 7427 . . . . 5 (𝑥 = 𝐶 → (𝐵o 𝑥) = (𝐵o 𝐶))
1210, 11sseq12d 4010 . . . 4 (𝑥 = 𝐶 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
13 onelon 6396 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 8543 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = 1o)
16 oe0 8543 . . . . . . 7 (𝐵 ∈ On → (𝐵o ∅) = 1o)
1716adantr 479 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵o ∅) = 1o)
1815, 17eqtr4d 2768 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = (𝐵o ∅))
19 eqimss 4035 . . . . 5 ((𝐴o ∅) = (𝐵o ∅) → (𝐴o ∅) ⊆ (𝐵o ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) ⊆ (𝐵o ∅))
21 simpl 481 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 6413 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 405 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 513 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 8558 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
26253adant2 1128 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
27 oecl 8558 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
28273adant1 1127 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
29 simp1 1133 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 8593 . . . . . . . . . . . . 13 (((𝐴o 𝑦) ∈ On ∧ (𝐵o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴)))
3126, 28, 29, 30syl3anc 1368 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴)))
3231imp 405 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
3332adantrl 714 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
34 omwordi 8592 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵o 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3528, 34syld3an3 1406 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3635imp 405 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3736adantrr 715 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3833, 37sstrd 3987 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
39 oesuc 8548 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
40393adant2 1128 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
4140adantr 479 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
42 oesuc 8548 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
43423adant1 1127 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
4443adantr 479 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
4538, 41, 443sstr4d 4024 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))
4645exp520 1354 . . . . . . 7 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴𝐵 → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))))))
4746com3r 87 . . . . . 6 (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))))))
4847imp4c 422 . . . . 5 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))))
4924, 48syl5 34 . . . 4 (𝑦 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))))
50 vex 3465 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 6435 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 688 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 6434 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 8542 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
5554biimpa 475 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
5652, 53, 55syl2anc 582 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57 0ss 4398 . . . . . . . . . 10 ∅ ⊆ (𝐵o 𝑥)
5856, 57eqsstrdi 4031 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥))
59 oveq1 7426 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴o 𝑥) = (∅ ↑o 𝑥))
6059sseq1d 4008 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥)))
6158, 60imbitrrid 245 . . . . . . . 8 (𝐴 = ∅ → (Lim 𝑥 → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
6261adantl 480 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
6362a1dd 50 . . . . . 6 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
64 ss2iun 5015 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦))
65 oelim 8555 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6650, 65mpanlr1 704 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6766an32s 650 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6867adantllr 717 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6921anim1i 613 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70 ne0i 4334 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 6427 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71imbitrrid 245 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 405 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 479 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 8555 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7650, 75mpanlr1 704 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7769, 74, 76syl2anc 582 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7877ad4ant24 752 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7968, 78sseq12d 4010 . . . . . . . 8 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦)))
8064, 79imbitrrid 245 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
8180ex 411 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8263, 81oe0lem 8534 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8313ancri 548 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)))
8482, 83syl11 33 . . . 4 (Lim 𝑥 → ((𝐵 ∈ On ∧ 𝐴𝐵) → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
853, 6, 9, 12, 20, 49, 84tfinds3 7870 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
8685expd 414 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶))))
8786impcom 406 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  wss 3944  c0 4322   ciun 4997  Oncon0 6371  Lim wlim 6372  suc csuc 6373  (class class class)co 7419  1oc1o 8480   ·o comu 8485  o coe 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-oexp 8493
This theorem is referenced by:  oeordsuc  8615  oege2  42878
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