Theorem ccatfval 13358

Description: Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
ccatfval	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))

Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ccatfval

Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		elex 3212	. 2 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		fveq2 6191	. . . . . 6 ⊢ (𝑠 = 𝑆 → (#‘𝑠) = (#‘𝑆))
4		fveq2 6191	. . . . . 6 ⊢ (𝑡 = 𝑇 → (#‘𝑡) = (#‘𝑇))
5	3, 4	oveqan12d 6669	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((#‘𝑠) + (#‘𝑡)) = ((#‘𝑆) + (#‘𝑇)))
6	5	oveq2d 6666	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (0..^((#‘𝑠) + (#‘𝑡))) = (0..^((#‘𝑆) + (#‘𝑇))))
7	3	oveq2d 6666	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0..^(#‘𝑠)) = (0..^(#‘𝑆)))
8	7	eleq2d 2687	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ (0..^(#‘𝑠)) ↔ 𝑥 ∈ (0..^(#‘𝑆))))
9	8	adantr 481	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 ∈ (0..^(#‘𝑠)) ↔ 𝑥 ∈ (0..^(#‘𝑆))))
10		fveq1 6190	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑥) = (𝑆‘𝑥))
11	10	adantr 481	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠‘𝑥) = (𝑆‘𝑥))
12		simpr 477	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
13	3	oveq2d 6666	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 − (#‘𝑠)) = (𝑥 − (#‘𝑆)))
14	13	adantr 481	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 − (#‘𝑠)) = (𝑥 − (#‘𝑆)))
15	12, 14	fveq12d 6197	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑡‘(𝑥 − (#‘𝑠))) = (𝑇‘(𝑥 − (#‘𝑆))))
16	9, 11, 15	ifbieq12d 4113	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (#‘𝑠)))) = if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))))
17	6, 16	mpteq12dv 4733	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (#‘𝑠))))) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))
18		df-concat 13301	. . 3 ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
19		ovex 6678	. . . 4 ⊢ (0..^((#‘𝑆) + (#‘𝑇))) ∈ V
20	19	mptex 6486	. . 3 ⊢ (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))) ∈ V
21	17, 18, 20	ovmpt2a 6791	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))
22	1, 2, 21	syl2an 494	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ifcif 4086   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  0cc0 9936   + caddc 9939   − cmin 10266  ..^cfzo 12465  #chash 13117   ++ cconcat 13293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-concat 13301

This theorem is referenced by:  ccatcl  13359  ccatlen  13360  ccatval1  13361  ccatval2  13362  ccatvalfn  13365  ccatalpha  13375  repswccat  13532  ccatco  13581  ofccat  13708  ccatmulgnn0dir  30619