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

Theorem nnaordi 7562
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 6944 . . . . . 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 6942 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
5 ordsucss 6887 . . . . . . . . 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 6950 . . . . . . . . . 10 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9 oveq2 6535 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝐴))
109sseq2d 3595 . . . . . . . . . . . . 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 6535 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝑦))
1312sseq2d 3595 . . . . . . . . . . . . 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 6535 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 suc 𝑦))
1615sseq2d 3595 . . . . . . . . . . . . 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 6535 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +𝑜 𝑥) = (𝐶 +𝑜 𝐵))
1918sseq2d 3595 . . . . . . . . . . . . 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 3586 . . . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . 17 (𝐶 +𝑜 𝑦) ⊆ suc (𝐶 +𝑜 𝑦)
24 sstr2 3574 . . . . . . . . . . . . . . . . 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 7550 . . . . . . . . . . . . . . . . . 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 3595 . . . . . . . . . . . . . . . 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 6962 . . . . . . . . . . 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 7550 . . . . . . . . . 10 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 suc 𝐴) = suc (𝐶 +𝑜 𝐴))
3938sseq1d 3594 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +𝑜 suc 𝐴) ⊆ (𝐶 +𝑜 𝐵) ↔ suc (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
40 ovex 6555 . . . . . . . . . 10 (𝐶 +𝑜 𝐴) ∈ V
41 sucssel 5722 . . . . . . . . . 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 1976  Vcvv 3172  wss 3539  Ord word 5625  suc csuc 5628  (class class class)co 6527  ωcom 6934   +𝑜 coa 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-oadd 7428
This theorem is referenced by:  nnaord  7563  nnmordi  7575  addclpi  9570  addnidpi  9579
  Copyright terms: Public domain W3C validator