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Theorem nnaordi 6017
Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaordi ((B 𝜔 𝐶 𝜔) → (A B → (𝐶 +𝑜 A) (𝐶 +𝑜 B)))

Proof of Theorem nnaordi
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5463 . . . . . . . . 9 (x = 𝐶 → (A +𝑜 x) = (A +𝑜 𝐶))
2 oveq2 5463 . . . . . . . . 9 (x = 𝐶 → (B +𝑜 x) = (B +𝑜 𝐶))
31, 2eleq12d 2105 . . . . . . . 8 (x = 𝐶 → ((A +𝑜 x) (B +𝑜 x) ↔ (A +𝑜 𝐶) (B +𝑜 𝐶)))
43imbi2d 219 . . . . . . 7 (x = 𝐶 → (((B 𝜔 A B) → (A +𝑜 x) (B +𝑜 x)) ↔ ((B 𝜔 A B) → (A +𝑜 𝐶) (B +𝑜 𝐶))))
5 oveq2 5463 . . . . . . . . 9 (x = ∅ → (A +𝑜 x) = (A +𝑜 ∅))
6 oveq2 5463 . . . . . . . . 9 (x = ∅ → (B +𝑜 x) = (B +𝑜 ∅))
75, 6eleq12d 2105 . . . . . . . 8 (x = ∅ → ((A +𝑜 x) (B +𝑜 x) ↔ (A +𝑜 ∅) (B +𝑜 ∅)))
8 oveq2 5463 . . . . . . . . 9 (x = y → (A +𝑜 x) = (A +𝑜 y))
9 oveq2 5463 . . . . . . . . 9 (x = y → (B +𝑜 x) = (B +𝑜 y))
108, 9eleq12d 2105 . . . . . . . 8 (x = y → ((A +𝑜 x) (B +𝑜 x) ↔ (A +𝑜 y) (B +𝑜 y)))
11 oveq2 5463 . . . . . . . . 9 (x = suc y → (A +𝑜 x) = (A +𝑜 suc y))
12 oveq2 5463 . . . . . . . . 9 (x = suc y → (B +𝑜 x) = (B +𝑜 suc y))
1311, 12eleq12d 2105 . . . . . . . 8 (x = suc y → ((A +𝑜 x) (B +𝑜 x) ↔ (A +𝑜 suc y) (B +𝑜 suc y)))
14 simpr 103 . . . . . . . . 9 ((B 𝜔 A B) → A B)
15 elnn 4271 . . . . . . . . . . 11 ((A B B 𝜔) → A 𝜔)
1615ancoms 255 . . . . . . . . . 10 ((B 𝜔 A B) → A 𝜔)
17 nna0 5992 . . . . . . . . . 10 (A 𝜔 → (A +𝑜 ∅) = A)
1816, 17syl 14 . . . . . . . . 9 ((B 𝜔 A B) → (A +𝑜 ∅) = A)
19 nna0 5992 . . . . . . . . . 10 (B 𝜔 → (B +𝑜 ∅) = B)
2019adantr 261 . . . . . . . . 9 ((B 𝜔 A B) → (B +𝑜 ∅) = B)
2114, 18, 203eltr4d 2118 . . . . . . . 8 ((B 𝜔 A B) → (A +𝑜 ∅) (B +𝑜 ∅))
22 simprl 483 . . . . . . . . . . . . 13 ((y 𝜔 (B 𝜔 A B)) → B 𝜔)
23 simpl 102 . . . . . . . . . . . . 13 ((y 𝜔 (B 𝜔 A B)) → y 𝜔)
24 nnacl 5998 . . . . . . . . . . . . 13 ((B 𝜔 y 𝜔) → (B +𝑜 y) 𝜔)
2522, 23, 24syl2anc 391 . . . . . . . . . . . 12 ((y 𝜔 (B 𝜔 A B)) → (B +𝑜 y) 𝜔)
26 nnsucelsuc 6009 . . . . . . . . . . . 12 ((B +𝑜 y) 𝜔 → ((A +𝑜 y) (B +𝑜 y) ↔ suc (A +𝑜 y) suc (B +𝑜 y)))
2725, 26syl 14 . . . . . . . . . . 11 ((y 𝜔 (B 𝜔 A B)) → ((A +𝑜 y) (B +𝑜 y) ↔ suc (A +𝑜 y) suc (B +𝑜 y)))
2816adantl 262 . . . . . . . . . . . . . 14 ((y 𝜔 (B 𝜔 A B)) → A 𝜔)
29 nnon 4275 . . . . . . . . . . . . . 14 (A 𝜔 → A On)
3028, 29syl 14 . . . . . . . . . . . . 13 ((y 𝜔 (B 𝜔 A B)) → A On)
31 nnon 4275 . . . . . . . . . . . . . 14 (y 𝜔 → y On)
3231adantr 261 . . . . . . . . . . . . 13 ((y 𝜔 (B 𝜔 A B)) → y On)
33 oasuc 5983 . . . . . . . . . . . . 13 ((A On y On) → (A +𝑜 suc y) = suc (A +𝑜 y))
3430, 32, 33syl2anc 391 . . . . . . . . . . . 12 ((y 𝜔 (B 𝜔 A B)) → (A +𝑜 suc y) = suc (A +𝑜 y))
35 nnon 4275 . . . . . . . . . . . . . 14 (B 𝜔 → B On)
3635ad2antrl 459 . . . . . . . . . . . . 13 ((y 𝜔 (B 𝜔 A B)) → B On)
37 oasuc 5983 . . . . . . . . . . . . 13 ((B On y On) → (B +𝑜 suc y) = suc (B +𝑜 y))
3836, 32, 37syl2anc 391 . . . . . . . . . . . 12 ((y 𝜔 (B 𝜔 A B)) → (B +𝑜 suc y) = suc (B +𝑜 y))
3934, 38eleq12d 2105 . . . . . . . . . . 11 ((y 𝜔 (B 𝜔 A B)) → ((A +𝑜 suc y) (B +𝑜 suc y) ↔ suc (A +𝑜 y) suc (B +𝑜 y)))
4027, 39bitr4d 180 . . . . . . . . . 10 ((y 𝜔 (B 𝜔 A B)) → ((A +𝑜 y) (B +𝑜 y) ↔ (A +𝑜 suc y) (B +𝑜 suc y)))
4140biimpd 132 . . . . . . . . 9 ((y 𝜔 (B 𝜔 A B)) → ((A +𝑜 y) (B +𝑜 y) → (A +𝑜 suc y) (B +𝑜 suc y)))
4241ex 108 . . . . . . . 8 (y 𝜔 → ((B 𝜔 A B) → ((A +𝑜 y) (B +𝑜 y) → (A +𝑜 suc y) (B +𝑜 suc y))))
437, 10, 13, 21, 42finds2 4267 . . . . . . 7 (x 𝜔 → ((B 𝜔 A B) → (A +𝑜 x) (B +𝑜 x)))
444, 43vtoclga 2613 . . . . . 6 (𝐶 𝜔 → ((B 𝜔 A B) → (A +𝑜 𝐶) (B +𝑜 𝐶)))
4544imp 115 . . . . 5 ((𝐶 𝜔 (B 𝜔 A B)) → (A +𝑜 𝐶) (B +𝑜 𝐶))
4616adantl 262 . . . . . 6 ((𝐶 𝜔 (B 𝜔 A B)) → A 𝜔)
47 simpl 102 . . . . . 6 ((𝐶 𝜔 (B 𝜔 A B)) → 𝐶 𝜔)
48 nnacom 6002 . . . . . 6 ((A 𝜔 𝐶 𝜔) → (A +𝑜 𝐶) = (𝐶 +𝑜 A))
4946, 47, 48syl2anc 391 . . . . 5 ((𝐶 𝜔 (B 𝜔 A B)) → (A +𝑜 𝐶) = (𝐶 +𝑜 A))
50 nnacom 6002 . . . . . . 7 ((B 𝜔 𝐶 𝜔) → (B +𝑜 𝐶) = (𝐶 +𝑜 B))
5150ancoms 255 . . . . . 6 ((𝐶 𝜔 B 𝜔) → (B +𝑜 𝐶) = (𝐶 +𝑜 B))
5251adantrr 448 . . . . 5 ((𝐶 𝜔 (B 𝜔 A B)) → (B +𝑜 𝐶) = (𝐶 +𝑜 B))
5345, 49, 523eltr3d 2117 . . . 4 ((𝐶 𝜔 (B 𝜔 A B)) → (𝐶 +𝑜 A) (𝐶 +𝑜 B))
54533impb 1099 . . 3 ((𝐶 𝜔 B 𝜔 A B) → (𝐶 +𝑜 A) (𝐶 +𝑜 B))
55543com12 1107 . 2 ((B 𝜔 𝐶 𝜔 A B) → (𝐶 +𝑜 A) (𝐶 +𝑜 B))
56553expia 1105 1 ((B 𝜔 𝐶 𝜔) → (A B → (𝐶 +𝑜 A) (𝐶 +𝑜 B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944
This theorem is referenced by:  nnaord  6018  nnmordi  6025  addclpi  6311  addnidpig  6320  archnqq  6400  prarloclemarch2  6402  prarloclemlt  6475
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