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

Theorem oeordsuc 7911
Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)
Assertion
Ref Expression
oeordsuc ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))

Proof of Theorem oeordsuc
StepHypRef Expression
1 onelon 5961 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 399 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 468 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 7909 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
543adant1 1153 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
6 oecl 7854 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
763adant2 1154 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
8 oecl 7854 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 𝐶) ∈ On)
983adant1 1153 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 𝐶) ∈ On)
10 simp1 1159 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 7889 . . . . . . . . . . 11 (((𝐴𝑜 𝐶) ∈ On ∧ (𝐵𝑜 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶) → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
127, 9, 10, 11syl3anc 1483 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶) → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
135, 12syld 47 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
14 oesuc 7844 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
15143adant2 1154 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
1615sseq1d 3829 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ↔ ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
1713, 16sylibrd 250 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
18 ne0i 4122 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 5993 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19syl5ibr 237 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 468 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 7903 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵𝑜 𝐶))
2322ex 399 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵𝑜 𝐶)))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ (𝐵𝑜 𝐶)))
25 simpl 470 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
2625, 8jca 503 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ On ∧ (𝐵𝑜 𝐶) ∈ On))
27 omordi 7883 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ (𝐵𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐵𝑜 𝐶)) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
2826, 27sylan 571 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵𝑜 𝐶)) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
2928ex 399 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵𝑜 𝐶) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵))))
3029com23 86 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (∅ ∈ (𝐵𝑜 𝐶) → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵))))
3124, 30mpdd 43 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
32313adant1 1153 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
33 oesuc 7844 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) = ((𝐵𝑜 𝐶) ·𝑜 𝐵))
34333adant1 1153 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) = ((𝐵𝑜 𝐶) ·𝑜 𝐵))
3534eleq2d 2871 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶) ↔ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
3632, 35sylibrd 250 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶)))
3717, 36jcad 504 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶))))
38373expa 1140 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶))))
39 sucelon 7247 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
40 oecl 7854 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) ∈ On)
41 oecl 7854 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) ∈ On)
42 ontr2 5985 . . . . . . . . 9 (((𝐴𝑜 suc 𝐶) ∈ On ∧ (𝐵𝑜 suc 𝐶) ∈ On) → (((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶)) → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4340, 41, 42syl2an 585 . . . . . . . 8 (((𝐴 ∈ On ∧ suc 𝐶 ∈ On) ∧ (𝐵 ∈ On ∧ suc 𝐶 ∈ On)) → (((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶)) → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4443anandirs 661 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc 𝐶 ∈ On) → (((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶)) → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4539, 44sylan2b 583 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶)) → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4638, 45syld 47 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4746exp31 408 . . . 4 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))))
4847com4l 92 . . 3 (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴 ∈ On → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))))
4948imp 395 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 ∈ On → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶))))
503, 49mpdd 43 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wss 3769  c0 4116  Oncon0 5936  suc csuc 5938  (class class class)co 6874   ·𝑜 comu 7794  𝑜 coe 7795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-omul 7801  df-oexp 7802
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator