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Theorem nnaordi 6503
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) → (𝐴𝐵 → (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 5877 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
2 oveq2 5877 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
31, 2eleq12d 2248 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶)))
43imbi2d 230 . . . . . . 7 (𝑥 = 𝐶 → (((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)) ↔ ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))))
5 oveq2 5877 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
6 oveq2 5877 . . . . . . . . 9 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
75, 6eleq12d 2248 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ∈ (𝐵 +o ∅)))
8 oveq2 5877 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
9 oveq2 5877 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
108, 9eleq12d 2248 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦)))
11 oveq2 5877 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12 oveq2 5877 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1311, 12eleq12d 2248 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
14 simpr 110 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴𝐵)
15 elnn 4602 . . . . . . . . . . 11 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
1615ancoms 268 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
17 nna0 6469 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
1816, 17syl 14 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
19 nna0 6469 . . . . . . . . . 10 (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
2019adantr 276 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵 +o ∅) = 𝐵)
2114, 18, 203eltr4d 2261 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) ∈ (𝐵 +o ∅))
22 simprl 529 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
23 simpl 109 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ ω)
24 nnacl 6475 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
2522, 23, 24syl2anc 411 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +o 𝑦) ∈ ω)
26 nnsucelsuc 6486 . . . . . . . . . . . 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 ∧ 𝐴𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ∈ suc (𝐵 +o 𝑦)))
2816adantl 277 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
29 nnon 4606 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → 𝐴 ∈ On)
3028, 29syl 14 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ On)
31 nnon 4606 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ On)
3231adantr 276 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ On)
33 oasuc 6459 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
3430, 32, 33syl2anc 411 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
35 nnon 4606 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → 𝐵 ∈ On)
3635ad2antrl 490 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ On)
37 oasuc 6459 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3836, 32, 37syl2anc 411 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3934, 38eleq12d 2248 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦) ↔ suc (𝐴 +o 𝑦) ∈ suc (𝐵 +o 𝑦)))
4027, 39bitr4d 191 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) ↔ (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
4140biimpd 144 . . . . . . . . 9 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
4241ex 115 . . . . . . . 8 (𝑦 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦))))
437, 10, 13, 21, 42finds2 4597 . . . . . . 7 (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)))
444, 43vtoclga 2803 . . . . . 6 (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶)))
4544imp 124 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))
4616adantl 277 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
47 simpl 109 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐶 ∈ ω)
48 nnacom 6479 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
4946, 47, 48syl2anc 411 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
50 nnacom 6479 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
5150ancoms 268 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
5251adantrr 479 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
5345, 49, 523eltr3d 2260 . . . 4 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
54533impb 1199 . . 3 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
55543com12 1207 . 2 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
56553expia 1205 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  c0 3422  Oncon0 4360  suc csuc 4362  ωcom 4586  (class class class)co 5869   +o coa 6408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415
This theorem is referenced by:  nnaord  6504  nnmordi  6511  addclpi  7317  addnidpig  7326  archnqq  7407  prarloclemarch2  7409  prarloclemlt  7483
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