MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oewordri Structured version   Visualization version   GIF version

Theorem oewordri 8473
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 7325 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
2 oveq2 7325 . . . . 5 (𝑥 = ∅ → (𝐵o 𝑥) = (𝐵o ∅))
31, 2sseq12d 3964 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o ∅) ⊆ (𝐵o ∅)))
4 oveq2 7325 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
5 oveq2 7325 . . . . 5 (𝑥 = 𝑦 → (𝐵o 𝑥) = (𝐵o 𝑦))
64, 5sseq12d 3964 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)))
7 oveq2 7325 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
8 oveq2 7325 . . . . 5 (𝑥 = suc 𝑦 → (𝐵o 𝑥) = (𝐵o suc 𝑦))
97, 8sseq12d 3964 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦)))
10 oveq2 7325 . . . . 5 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
11 oveq2 7325 . . . . 5 (𝑥 = 𝐶 → (𝐵o 𝑥) = (𝐵o 𝐶))
1210, 11sseq12d 3964 . . . 4 (𝑥 = 𝐶 → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
13 onelon 6314 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
14 oe0 8402 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1513, 14syl 17 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = 1o)
16 oe0 8402 . . . . . . 7 (𝐵 ∈ On → (𝐵o ∅) = 1o)
1716adantr 481 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐵o ∅) = 1o)
1815, 17eqtr4d 2780 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) = (𝐵o ∅))
19 eqimss 3987 . . . . 5 ((𝐴o ∅) = (𝐵o ∅) → (𝐴o ∅) ⊆ (𝐵o ∅))
2018, 19syl 17 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o ∅) ⊆ (𝐵o ∅))
21 simpl 483 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ On)
22 onelss 6331 . . . . . . 7 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
2322imp 407 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝐵)
2413, 21, 23jca31 515 . . . . 5 ((𝐵 ∈ On ∧ 𝐴𝐵) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵))
25 oecl 8417 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
26253adant2 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
27 oecl 8417 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
28273adant1 1129 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o 𝑦) ∈ On)
29 simp1 1135 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
30 omwordri 8453 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝑦) ⊆ (𝐵o 𝑦) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴)))
3231imp 407 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦)) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
3332adantrl 713 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐴))
34 omwordi 8452 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵o 𝑦) ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3528, 34syld3an3 1408 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵)))
3635imp 407 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3736adantrr 714 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐵o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
3833, 37sstrd 3941 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → ((𝐴o 𝑦) ·o 𝐴) ⊆ ((𝐵o 𝑦) ·o 𝐵))
39 oesuc 8407 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
40393adant2 1130 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
4140adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
42 oesuc 8407 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
43423adant1 1129 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
4443adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐵o suc 𝑦) = ((𝐵o 𝑦) ·o 𝐵))
4538, 41, 443sstr4d 3978 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴o 𝑦) ⊆ (𝐵o 𝑦))) → (𝐴o suc 𝑦) ⊆ (𝐵o suc 𝑦))
4645exp520 1356 . . . . . . 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 424 . . . . 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 3445 . . . . . . . . . . . 12 𝑥 ∈ V
51 limelon 6352 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5250, 51mpan 687 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
53 0ellim 6351 . . . . . . . . . . 11 (Lim 𝑥 → ∅ ∈ 𝑥)
54 oe0m1 8401 . . . . . . . . . . . 12 (𝑥 ∈ On → (∅ ∈ 𝑥 ↔ (∅ ↑o 𝑥) = ∅))
5554biimpa 477 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → (∅ ↑o 𝑥) = ∅)
5652, 53, 55syl2anc 584 . . . . . . . . . 10 (Lim 𝑥 → (∅ ↑o 𝑥) = ∅)
57 0ss 4341 . . . . . . . . . 10 ∅ ⊆ (𝐵o 𝑥)
5856, 57eqsstrdi 3985 . . . . . . . . 9 (Lim 𝑥 → (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥))
59 oveq1 7324 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴o 𝑥) = (∅ ↑o 𝑥))
6059sseq1d 3962 . . . . . . . . 9 (𝐴 = ∅ → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ (∅ ↑o 𝑥) ⊆ (𝐵o 𝑥)))
6158, 60syl5ibr 245 . . . . . . . 8 (𝐴 = ∅ → (Lim 𝑥 → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
6261adantl 482 . . . . . . 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 4955 . . . . . . . 8 (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦))
65 oelim 8414 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6650, 65mpanlr1 703 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6766an32s 649 . . . . . . . . . 10 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6867adantllr 716 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
6921anim1i 615 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵 ∈ On ∧ Lim 𝑥))
70 ne0i 4279 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
71 on0eln0 6344 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
7270, 71syl5ibr 245 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
7372imp 407 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴𝐵) → ∅ ∈ 𝐵)
7473adantr 481 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → ∅ ∈ 𝐵)
75 oelim 8414 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7650, 75mpanlr1 703 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7769, 74, 76syl2anc 584 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐴𝐵) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7877ad4ant24 751 . . . . . . . . 9 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (𝐵o 𝑥) = 𝑦𝑥 (𝐵o 𝑦))
7968, 78sseq12d 3964 . . . . . . . 8 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → ((𝐴o 𝑥) ⊆ (𝐵o 𝑥) ↔ 𝑦𝑥 (𝐴o 𝑦) ⊆ 𝑦𝑥 (𝐵o 𝑦)))
8064, 79syl5ibr 245 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥)))
8180ex 413 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8263, 81oe0lem 8393 . . . . 5 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐴𝐵)) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴o 𝑦) ⊆ (𝐵o 𝑦) → (𝐴o 𝑥) ⊆ (𝐵o 𝑥))))
8313ancri 550 . . . . 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 7758 . . 3 (𝐶 ∈ On → ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
8685expd 416 . 2 (𝐶 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶))))
8786impcom 408 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2941  wral 3062  Vcvv 3441  wss 3897  c0 4267   ciun 4937  Oncon0 6289  Lim wlim 6290  suc csuc 6291  (class class class)co 7317  1oc1o 8339   ·o comu 8344  o coe 8345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7630
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-oadd 8350  df-omul 8351  df-oexp 8352
This theorem is referenced by:  oeordsuc  8475
  Copyright terms: Public domain W3C validator