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Theorem nnaordi 7465
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 elnn 6848 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21ancoms 467 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
32adantll 745 . . . 4 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
4 nnord 6846 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
5 ordsucss 6791 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
64, 5syl 17 . . . . . . . 8 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴𝐵))
76ad2antlr 758 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴𝐵 → suc 𝐴𝐵))
8 peano2b 6854 . . . . . . . . . 10 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9 oveq2 6439 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝐴))
109sseq2d 3500 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)))
1110imbi2d 328 . . . . . . . . . . . 12 (𝑥 = suc 𝐴 → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴))))
12 oveq2 6439 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝑦))
1312sseq2d 3500 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦)))
1413imbi2d 328 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦))))
15 oveq2 6439 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝑦))
1615sseq2d 3500 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
1716imbi2d 328 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
18 oveq2 6439 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝐵))
1918sseq2d 3500 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
2019imbi2d 328 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
21 ssid 3491 . . . . . . . . . . . . 13 (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)
22212a1i 12 . . . . . . . . . . . 12 (suc 𝐴 ∈ ω → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)))
23 sssucid 5609 . . . . . . . . . . . . . . . . 17 (𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦)
24 sstr2 3479 . . . . . . . . . . . . . . . . 17 ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → ((𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦)))
2523, 24mpi 20 . . . . . . . . . . . . . . . 16 ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦))
26 nnasuc 7453 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2726ancoms 467 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2827sseq2d 3500 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦)))
2925, 28syl5ibr 234 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
3029ex 448 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3130ad2antrr 757 . . . . . . . . . . . . 13 (((𝑦 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑦) → (𝐶 ∈ ω → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3231a2d 29 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝑦) → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦)) → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3311, 14, 17, 20, 22, 32findsg 6866 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝐵) → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
3433exp31 627 . . . . . . . . . 10 (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
358, 34syl5bi 230 . . . . . . . . 9 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
3635com4r 91 . . . . . . . 8 (𝐶 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
3736imp31 446 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
38 nnasuc 7453 . . . . . . . . . 10 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 suc 𝐴) = suc (𝐶 +𝑜 𝐴))
3938sseq1d 3499 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) ↔ suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
40 ovex 6459 . . . . . . . . . 10 (𝐶 +𝑜 𝐴) ∈ V
41 sucssel 5626 . . . . . . . . . 10 ((𝐶 +𝑜 𝐴) ∈ V → (suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
4240, 41ax-mp 5 . . . . . . . . 9 (suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
4339, 42syl6bi 241 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
4443adantlr 746 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
457, 37, 443syld 57 . . . . . 6 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
4645imp 443 . . . . 5 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
4746an32s 841 . . . 4 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
483, 47mpdan 698 . . 3 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
4948ex 448 . 2 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
5049ancoms 467 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  wss 3444  Ord word 5529  suc csuc 5532  (class class class)co 6431  ωcom 6838   +𝑜 coa 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-ov 6434  df-oprab 6435  df-mpt2 6436  df-om 6839  df-wrecs 7174  df-recs 7235  df-rdg 7273  df-oadd 7331
This theorem is referenced by:  nnaord  7466  nnmordi  7478  addclpi  9473  addnidpi  9482
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