ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  smoel Structured version   GIF version

Theorem smoel 5856
Description: If x is less than y then a strictly monotone function's value will be strictly less at x than at y. (Contributed by Andrew Salmon, 22-Nov-2011.)
Assertion
Ref Expression
smoel ((Smo B A dom B 𝐶 A) → (B𝐶) (BA))

Proof of Theorem smoel
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smodm 5847 . . . . 5 (Smo B → Ord dom B)
2 ordtr1 4091 . . . . . . 7 (Ord dom B → ((𝐶 A A dom B) → 𝐶 dom B))
32ancomsd 256 . . . . . 6 (Ord dom B → ((A dom B 𝐶 A) → 𝐶 dom B))
43expdimp 246 . . . . 5 ((Ord dom B A dom B) → (𝐶 A𝐶 dom B))
51, 4sylan 267 . . . 4 ((Smo B A dom B) → (𝐶 A𝐶 dom B))
6 df-smo 5842 . . . . . 6 (Smo B ↔ (B:dom B⟶On Ord dom B x dom By dom B(x y → (Bx) (By))))
7 eleq1 2097 . . . . . . . . . . 11 (x = 𝐶 → (x y𝐶 y))
8 fveq2 5121 . . . . . . . . . . . 12 (x = 𝐶 → (Bx) = (B𝐶))
98eleq1d 2103 . . . . . . . . . . 11 (x = 𝐶 → ((Bx) (By) ↔ (B𝐶) (By)))
107, 9imbi12d 223 . . . . . . . . . 10 (x = 𝐶 → ((x y → (Bx) (By)) ↔ (𝐶 y → (B𝐶) (By))))
11 eleq2 2098 . . . . . . . . . . 11 (y = A → (𝐶 y𝐶 A))
12 fveq2 5121 . . . . . . . . . . . 12 (y = A → (By) = (BA))
1312eleq2d 2104 . . . . . . . . . . 11 (y = A → ((B𝐶) (By) ↔ (B𝐶) (BA)))
1411, 13imbi12d 223 . . . . . . . . . 10 (y = A → ((𝐶 y → (B𝐶) (By)) ↔ (𝐶 A → (B𝐶) (BA))))
1510, 14rspc2v 2656 . . . . . . . . 9 ((𝐶 dom B A dom B) → (x dom By dom B(x y → (Bx) (By)) → (𝐶 A → (B𝐶) (BA))))
1615ancoms 255 . . . . . . . 8 ((A dom B 𝐶 dom B) → (x dom By dom B(x y → (Bx) (By)) → (𝐶 A → (B𝐶) (BA))))
1716com12 27 . . . . . . 7 (x dom By dom B(x y → (Bx) (By)) → ((A dom B 𝐶 dom B) → (𝐶 A → (B𝐶) (BA))))
18173ad2ant3 926 . . . . . 6 ((B:dom B⟶On Ord dom B x dom By dom B(x y → (Bx) (By))) → ((A dom B 𝐶 dom B) → (𝐶 A → (B𝐶) (BA))))
196, 18sylbi 114 . . . . 5 (Smo B → ((A dom B 𝐶 dom B) → (𝐶 A → (B𝐶) (BA))))
2019expdimp 246 . . . 4 ((Smo B A dom B) → (𝐶 dom B → (𝐶 A → (B𝐶) (BA))))
215, 20syld 40 . . 3 ((Smo B A dom B) → (𝐶 A → (𝐶 A → (B𝐶) (BA))))
2221pm2.43d 44 . 2 ((Smo B A dom B) → (𝐶 A → (B𝐶) (BA)))
23223impia 1100 1 ((Smo B A dom B 𝐶 A) → (B𝐶) (BA))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wral 2300  Ord word 4065  Oncon0 4066  dom cdm 4288  wf 4841  cfv 4845  Smo wsmo 5841
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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-tr 3846  df-iord 4069  df-iota 4810  df-fv 4853  df-smo 5842
This theorem is referenced by:  smoiun  5857  smoel2  5859
  Copyright terms: Public domain W3C validator