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Theorem nnaordi 7984
 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 7355 . . . . . 6 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
21ancoms 452 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
32adantll 704 . . . 4 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
4 nnord 7353 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
5 ordsucss 7298 . . . . . . . . 9 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
64, 5syl 17 . . . . . . . 8 (𝐵 ∈ ω → (𝐴𝐵 → suc 𝐴𝐵))
76ad2antlr 717 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴𝐵 → suc 𝐴𝐵))
8 peano2b 7361 . . . . . . . . . 10 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9 oveq2 6932 . . . . . . . . . . . . . 14 (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
109sseq2d 3852 . . . . . . . . . . . . 13 (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
1110imbi2d 332 . . . . . . . . . . . 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 6932 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
1312sseq2d 3852 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
1413imbi2d 332 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
15 oveq2 6932 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
1615sseq2d 3852 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
1716imbi2d 332 . . . . . . . . . . . 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 6932 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
1918sseq2d 3852 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
2019imbi2d 332 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
21 ssid 3842 . . . . . . . . . . . . 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 6055 . . . . . . . . . . . . . . . . 17 (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
24 sstr2 3828 . . . . . . . . . . . . . . . . 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 7972 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2726ancoms 452 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
2827sseq2d 3852 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
2925, 28syl5ibr 238 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
3029ex 403 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
3130ad2antrr 716 . . . . . . . . . . . . 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 7373 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴𝐵) → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
3433exp31 412 . . . . . . . . . 10 (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
358, 34syl5bi 234 . . . . . . . . 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 410 . . . . . . 7 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
38 nnasuc 7972 . . . . . . . . . 10 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
3938sseq1d 3851 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
40 ovex 6956 . . . . . . . . . 10 (𝐶 +o 𝐴) ∈ V
41 sucssel 6070 . . . . . . . . . 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 245 . . . . . . . 8 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
4443adantlr 705 . . . . . . 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 397 . . . . 5 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
4746an32s 642 . . . 4 ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
483, 47mpdan 677 . . 3 (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
4948ex 403 . 2 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
5049ancoms 452 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  Ord word 5977  suc csuc 5980  (class class class)co 6924  ωcom 7345   +o coa 7842 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-oadd 7849 This theorem is referenced by:  nnaord  7985  nnmordi  7997  addclpi  10051  addnidpi  10060
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