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Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnaword Structured version   GIF version

Theorem nnaword 5991
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 5439 . . . . . . 7 (x = 𝐶 → (x +𝑜 A) = (𝐶 +𝑜 A))
2 oveq1 5439 . . . . . . 7 (x = 𝐶 → (x +𝑜 B) = (𝐶 +𝑜 B))
31, 2sseq12d 2947 . . . . . 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 5439 . . . . . . 7 (x = ∅ → (x +𝑜 A) = (∅ +𝑜 A))
7 oveq1 5439 . . . . . . 7 (x = ∅ → (x +𝑜 B) = (∅ +𝑜 B))
86, 7sseq12d 2947 . . . . . 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 5439 . . . . . . 7 (x = y → (x +𝑜 A) = (y +𝑜 A))
11 oveq1 5439 . . . . . . 7 (x = y → (x +𝑜 B) = (y +𝑜 B))
1210, 11sseq12d 2947 . . . . . 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 5439 . . . . . . 7 (x = suc y → (x +𝑜 A) = (suc y +𝑜 A))
15 oveq1 5439 . . . . . . 7 (x = suc y → (x +𝑜 B) = (suc y +𝑜 B))
1614, 15sseq12d 2947 . . . . . 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 5968 . . . . . . . 8 (A 𝜔 → (∅ +𝑜 A) = A)
1918eqcomd 2023 . . . . . . 7 (A 𝜔 → A = (∅ +𝑜 A))
2019adantr 261 . . . . . 6 ((A 𝜔 B 𝜔) → A = (∅ +𝑜 A))
21 nna0r 5968 . . . . . . . 8 (B 𝜔 → (∅ +𝑜 B) = B)
2221eqcomd 2023 . . . . . . 7 (B 𝜔 → B = (∅ +𝑜 B))
2322adantl 262 . . . . . 6 ((A 𝜔 B 𝜔) → B = (∅ +𝑜 B))
2420, 23sseq12d 2947 . . . . 5 ((A 𝜔 B 𝜔) → (AB ↔ (∅ +𝑜 A) ⊆ (∅ +𝑜 B)))
25 nnacl 5970 . . . . . . . . . . 11 ((y 𝜔 A 𝜔) → (y +𝑜 A) 𝜔)
26253adant3 910 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → (y +𝑜 A) 𝜔)
27 nnacl 5970 . . . . . . . . . . 11 ((y 𝜔 B 𝜔) → (y +𝑜 B) 𝜔)
28273adant2 909 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → (y +𝑜 B) 𝜔)
29 nnsucsssuc 5982 . . . . . . . . . 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 393 . . . . . . . . 9 ((y 𝜔 A 𝜔 B 𝜔) → ((y +𝑜 A) ⊆ (y +𝑜 B) ↔ suc (y +𝑜 A) ⊆ suc (y +𝑜 B)))
31 nnasuc 5966 . . . . . . . . . . . . 13 ((A 𝜔 y 𝜔) → (A +𝑜 suc y) = suc (A +𝑜 y))
32 peano2 4241 . . . . . . . . . . . . . 14 (y 𝜔 → suc y 𝜔)
33 nnacom 5974 . . . . . . . . . . . . . 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 5974 . . . . . . . . . . . . . 14 ((A 𝜔 y 𝜔) → (A +𝑜 y) = (y +𝑜 A))
36 suceq 4084 . . . . . . . . . . . . . 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 2059 . . . . . . . . . . . 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 910 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → suc (y +𝑜 A) = (suc y +𝑜 A))
41 nnasuc 5966 . . . . . . . . . . . . 13 ((B 𝜔 y 𝜔) → (B +𝑜 suc y) = suc (B +𝑜 y))
42 nnacom 5974 . . . . . . . . . . . . . 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 5974 . . . . . . . . . . . . . 14 ((B 𝜔 y 𝜔) → (B +𝑜 y) = (y +𝑜 B))
45 suceq 4084 . . . . . . . . . . . . . 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 2059 . . . . . . . . . . . 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 909 . . . . . . . . . 10 ((y 𝜔 A 𝜔 B 𝜔) → suc (y +𝑜 B) = (suc y +𝑜 B))
5040, 49sseq12d 2947 . . . . . . . . 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 1091 . . . . 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 4247 . . . 4 (x 𝜔 → ((A 𝜔 B 𝜔) → (AB ↔ (x +𝑜 A) ⊆ (x +𝑜 B))))
565, 55vtoclga 2592 . . 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 1083 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 871   = wceq 1226   wcel 1370  wss 2890  c0 3197  suc csuc 4047  𝜔com 4236  (class class class)co 5432   +𝑜 coa 5909
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-oadd 5916
This theorem is referenced by:  nnacan  5992  nnawordi  5995
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