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Theorem nnaword 6020
Description: Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaword ((A 𝜔 B 𝜔 𝐶 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))

Proof of Theorem nnaword
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5462 . . . . . . 7 (x = 𝐶 → (x +𝑜 A) = (𝐶 +𝑜 A))
2 oveq1 5462 . . . . . . 7 (x = 𝐶 → (x +𝑜 B) = (𝐶 +𝑜 B))
31, 2sseq12d 2968 . . . . . 6 (x = 𝐶 → ((x +𝑜 A) ⊆ (x +𝑜 B) ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))
43bibi2d 221 . . . . 5 (x = 𝐶 → ((AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B)) ↔ (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B))))
54imbi2d 219 . . . 4 (x = 𝐶 → (((A 𝜔 B 𝜔) → (AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B))) ↔ ((A 𝜔 B 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))))
6 oveq1 5462 . . . . . . 7 (x = ∅ → (x +𝑜 A) = (∅ +𝑜 A))
7 oveq1 5462 . . . . . . 7 (x = ∅ → (x +𝑜 B) = (∅ +𝑜 B))
86, 7sseq12d 2968 . . . . . 6 (x = ∅ → ((x +𝑜 A) ⊆ (x +𝑜 B) ↔ (∅ +𝑜 A) ⊆ (∅ +𝑜 B)))
98bibi2d 221 . . . . 5 (x = ∅ → ((AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B)) ↔ (AB ↔ (∅ +𝑜 A) ⊆ (∅ +𝑜 B))))
10 oveq1 5462 . . . . . . 7 (x = y → (x +𝑜 A) = (y +𝑜 A))
11 oveq1 5462 . . . . . . 7 (x = y → (x +𝑜 B) = (y +𝑜 B))
1210, 11sseq12d 2968 . . . . . 6 (x = y → ((x +𝑜 A) ⊆ (x +𝑜 B) ↔ (y +𝑜 A) ⊆ (y +𝑜 B)))
1312bibi2d 221 . . . . 5 (x = y → ((AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B)) ↔ (AB ↔ (y +𝑜 A) ⊆ (y +𝑜 B))))
14 oveq1 5462 . . . . . . 7 (x = suc y → (x +𝑜 A) = (suc y +𝑜 A))
15 oveq1 5462 . . . . . . 7 (x = suc y → (x +𝑜 B) = (suc y +𝑜 B))
1614, 15sseq12d 2968 . . . . . 6 (x = suc y → ((x +𝑜 A) ⊆ (x +𝑜 B) ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B)))
1716bibi2d 221 . . . . 5 (x = suc y → ((AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B)) ↔ (AB ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B))))
18 nna0r 5996 . . . . . . . 8 (A 𝜔 → (∅ +𝑜 A) = A)
1918eqcomd 2042 . . . . . . 7 (A 𝜔 → A = (∅ +𝑜 A))
2019adantr 261 . . . . . 6 ((A 𝜔 B 𝜔) → A = (∅ +𝑜 A))
21 nna0r 5996 . . . . . . . 8 (B 𝜔 → (∅ +𝑜 B) = B)
2221eqcomd 2042 . . . . . . 7 (B 𝜔 → B = (∅ +𝑜 B))
2322adantl 262 . . . . . 6 ((A 𝜔 B 𝜔) → B = (∅ +𝑜 B))
2420, 23sseq12d 2968 . . . . 5 ((A 𝜔 B 𝜔) → (AB ↔ (∅ +𝑜 A) ⊆ (∅ +𝑜 B)))
25 nnacl 5998 . . . . . . . . . . 11 ((y 𝜔 A 𝜔) → (y +𝑜 A) 𝜔)
26253adant3 923 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → (y +𝑜 A) 𝜔)
27 nnacl 5998 . . . . . . . . . . 11 ((y 𝜔 B 𝜔) → (y +𝑜 B) 𝜔)
28273adant2 922 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → (y +𝑜 B) 𝜔)
29 nnsucsssuc 6010 . . . . . . . . . 10 (((y +𝑜 A) 𝜔 (y +𝑜 B) 𝜔) → ((y +𝑜 A) ⊆ (y +𝑜 B) ↔ suc (y +𝑜 A) ⊆ suc (y +𝑜 B)))
3026, 28, 29syl2anc 391 . . . . . . . . 9 ((y 𝜔 A 𝜔 B 𝜔) → ((y +𝑜 A) ⊆ (y +𝑜 B) ↔ suc (y +𝑜 A) ⊆ suc (y +𝑜 B)))
31 nnasuc 5994 . . . . . . . . . . . . 13 ((A 𝜔 y 𝜔) → (A +𝑜 suc y) = suc (A +𝑜 y))
32 peano2 4261 . . . . . . . . . . . . . 14 (y 𝜔 → suc y 𝜔)
33 nnacom 6002 . . . . . . . . . . . . . 14 ((A 𝜔 suc y 𝜔) → (A +𝑜 suc y) = (suc y +𝑜 A))
3432, 33sylan2 270 . . . . . . . . . . . . 13 ((A 𝜔 y 𝜔) → (A +𝑜 suc y) = (suc y +𝑜 A))
35 nnacom 6002 . . . . . . . . . . . . . 14 ((A 𝜔 y 𝜔) → (A +𝑜 y) = (y +𝑜 A))
36 suceq 4105 . . . . . . . . . . . . . 14 ((A +𝑜 y) = (y +𝑜 A) → suc (A +𝑜 y) = suc (y +𝑜 A))
3735, 36syl 14 . . . . . . . . . . . . 13 ((A 𝜔 y 𝜔) → suc (A +𝑜 y) = suc (y +𝑜 A))
3831, 34, 373eqtr3rd 2078 . . . . . . . . . . . 12 ((A 𝜔 y 𝜔) → suc (y +𝑜 A) = (suc y +𝑜 A))
3938ancoms 255 . . . . . . . . . . 11 ((y 𝜔 A 𝜔) → suc (y +𝑜 A) = (suc y +𝑜 A))
40393adant3 923 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → suc (y +𝑜 A) = (suc y +𝑜 A))
41 nnasuc 5994 . . . . . . . . . . . . 13 ((B 𝜔 y 𝜔) → (B +𝑜 suc y) = suc (B +𝑜 y))
42 nnacom 6002 . . . . . . . . . . . . . 14 ((B 𝜔 suc y 𝜔) → (B +𝑜 suc y) = (suc y +𝑜 B))
4332, 42sylan2 270 . . . . . . . . . . . . 13 ((B 𝜔 y 𝜔) → (B +𝑜 suc y) = (suc y +𝑜 B))
44 nnacom 6002 . . . . . . . . . . . . . 14 ((B 𝜔 y 𝜔) → (B +𝑜 y) = (y +𝑜 B))
45 suceq 4105 . . . . . . . . . . . . . 14 ((B +𝑜 y) = (y +𝑜 B) → suc (B +𝑜 y) = suc (y +𝑜 B))
4644, 45syl 14 . . . . . . . . . . . . 13 ((B 𝜔 y 𝜔) → suc (B +𝑜 y) = suc (y +𝑜 B))
4741, 43, 463eqtr3rd 2078 . . . . . . . . . . . 12 ((B 𝜔 y 𝜔) → suc (y +𝑜 B) = (suc y +𝑜 B))
4847ancoms 255 . . . . . . . . . . 11 ((y 𝜔 B 𝜔) → suc (y +𝑜 B) = (suc y +𝑜 B))
49483adant2 922 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → suc (y +𝑜 B) = (suc y +𝑜 B))
5040, 49sseq12d 2968 . . . . . . . . 9 ((y 𝜔 A 𝜔 B 𝜔) → (suc (y +𝑜 A) ⊆ suc (y +𝑜 B) ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B)))
5130, 50bitrd 177 . . . . . . . 8 ((y 𝜔 A 𝜔 B 𝜔) → ((y +𝑜 A) ⊆ (y +𝑜 B) ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B)))
5251bibi2d 221 . . . . . . 7 ((y 𝜔 A 𝜔 B 𝜔) → ((AB ↔ (y +𝑜 A) ⊆ (y +𝑜 B)) ↔ (AB ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B))))
5352biimpd 132 . . . . . 6 ((y 𝜔 A 𝜔 B 𝜔) → ((AB ↔ (y +𝑜 A) ⊆ (y +𝑜 B)) → (AB ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B))))
54533expib 1106 . . . . 5 (y 𝜔 → ((A 𝜔 B 𝜔) → ((AB ↔ (y +𝑜 A) ⊆ (y +𝑜 B)) → (AB ↔ (suc y +𝑜 A) ⊆ (suc y +𝑜 B)))))
559, 13, 17, 24, 54finds2 4267 . . . 4 (x 𝜔 → ((A 𝜔 B 𝜔) → (AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B))))
565, 55vtoclga 2613 . . 3 (𝐶 𝜔 → ((A 𝜔 B 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B))))
5756impcom 116 . 2 (((A 𝜔 B 𝜔) 𝐶 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))
58573impa 1098 1 ((A 𝜔 B 𝜔 𝐶 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wss 2911  c0 3218  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944
This theorem is referenced by:  nnacan  6021  nnawordi  6024
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