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Theorem nnaordi 8233
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 7579 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21ancoms 459 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
32adantll 710 . . . 4 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
4 nnord 7577 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
5 ordsucss 7522 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
64, 5syl 17 . . . . . . . 8 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴𝐵))
76ad2antlr 723 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴𝐵 → suc 𝐴𝐵))
8 peano2b 7585 . . . . . . . . . 10 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9 oveq2 7153 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
109sseq2d 3996 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
1110imbi2d 342 . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
1312sseq2d 3996 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
1413imbi2d 342 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
15 oveq2 7153 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
1615sseq2d 3996 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
1716imbi2d 342 . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
1918sseq2d 3996 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
2019imbi2d 342 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
21 ssid 3986 . . . . . . . . . . . . 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 6261 . . . . . . . . . . . . . . . . 17 (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
24 sstr2 3971 . . . . . . . . . . . . . . . . 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 8221 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2726ancoms 459 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2827sseq2d 3996 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
2925, 28syl5ibr 247 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
3029ex 413 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
3130ad2antrr 722 . . . . . . . . . . . . 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 7598 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝐵) → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
3433exp31 420 . . . . . . . . . 10 (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
358, 34syl5bi 243 . . . . . . . . 9 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
3635com4r 94 . . . . . . . 8 (𝐶 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
3736imp31 418 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
38 nnasuc 8221 . . . . . . . . . 10 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
3938sseq1d 3995 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
40 ovex 7178 . . . . . . . . . 10 (𝐶 +o 𝐴) ∈ V
41 sucssel 6276 . . . . . . . . . 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 254 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
4443adantlr 711 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
457, 37, 443syld 60 . . . . . 6 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
4645imp 407 . . . . 5 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
4746an32s 648 . . . 4 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
483, 47mpdan 683 . . 3 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
4948ex 413 . 2 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
5049ancoms 459 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  Ord word 6183  suc csuc 6186  (class class class)co 7145  ωcom 7569   +o coa 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095
This theorem is referenced by:  nnaord  8234  nnmordi  8246  addclpi  10302  addnidpi  10311
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