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Theorem nnaordi 7396
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 6779 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21ancoms 462 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
32adantll 737 . . . 4 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
4 nnord 6777 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
5 ordsucss 6722 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
64, 5syl 17 . . . . . . . 8 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴𝐵))
76ad2antlr 750 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴𝐵 → suc 𝐴𝐵))
8 peano2b 6785 . . . . . . . . . 10 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9 oveq2 6371 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝐴))
109sseq2d 3482 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝐴)))
1110imbi2d 325 . . . . . . . . . . . 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 6371 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝑦))
1312sseq2d 3482 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦)))
1413imbi2d 325 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦))))
15 oveq2 6371 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝑦))
1615sseq2d 3482 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
1716imbi2d 325 . . . . . . . . . . . 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 6371 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝐵))
1918sseq2d 3482 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
2019imbi2d 325 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
21 ssid 3473 . . . . . . . . . . . . 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 5551 . . . . . . . . . . . . . . . . 17 (𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦)
24 sstr2 3461 . . . . . . . . . . . . . . . . 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 7384 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2726ancoms 462 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +𝑜 suc 𝑦) = suc (𝐶 +𝑜 𝑦))
2827sseq2d 3482 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦) ↔ (𝐶 +𝑜 suc 𝐴) ⊆ suc (𝐶 +𝑜 𝑦)))
2925, 28syl5ibr 231 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦)))
3029ex 443 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝑦) → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 suc 𝑦))))
3130ad2antrr 749 . . . . . . . . . . . . 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 6797 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝐵) → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
3433exp31 621 . . . . . . . . . 10 (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
358, 34syl5bi 227 . . . . . . . . 9 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
3635com4r 90 . . . . . . . 8 (𝐶 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
3736imp31 441 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴𝐵 → (𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
38 nnasuc 7384 . . . . . . . . . 10 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 suc 𝐴) = suc (𝐶 +𝑜 𝐴))
3938sseq1d 3481 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) ↔ suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
40 ovex 6391 . . . . . . . . . 10 (𝐶 +𝑜 𝐴) ∈ V
41 sucssel 5566 . . . . . . . . . 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 238 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
4443adantlr 738 . . . . . . 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 438 . . . . 5 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
4746an32s 832 . . . 4 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
483, 47mpdan 690 . . 3 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
4948ex 443 . 2 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
5049ancoms 462 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 378   = wceq 1468  wcel 1937  Vcvv 3066  wss 3426  Ord word 5473  suc csuc 5476  (class class class)co 6363  ωcom 6769   +𝑜 coa 7256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-oadd 7263
This theorem is referenced by:  nnaord  7397  nnmordi  7409  addclpi  9402  addnidpi  9411
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