Theorem ccatfval 13211

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 3185	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		elex 3185	. 2 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		fveq2 6103	. . . . . 6 ⊢ (𝑠 = 𝑆 → (#‘𝑠) = (#‘𝑆))
4		fveq2 6103	. . . . . 6 ⊢ (𝑡 = 𝑇 → (#‘𝑡) = (#‘𝑇))
5	3, 4	oveqan12d 6568	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((#‘𝑠) + (#‘𝑡)) = ((#‘𝑆) + (#‘𝑇)))
6	5	oveq2d 6565	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (0..^((#‘𝑠) + (#‘𝑡))) = (0..^((#‘𝑆) + (#‘𝑇))))
7	3	oveq2d 6565	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (0..^(#‘𝑠)) = (0..^(#‘𝑆)))
8	7	eleq2d 2673	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ (0..^(#‘𝑠)) ↔ 𝑥 ∈ (0..^(#‘𝑆))))
9	8	adantr 480	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 ∈ (0..^(#‘𝑠)) ↔ 𝑥 ∈ (0..^(#‘𝑆))))
10		fveq1 6102	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑥) = (𝑆‘𝑥))
11	10	adantr 480	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠‘𝑥) = (𝑆‘𝑥))
12		simpr 476	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
13	3	oveq2d 6565	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 − (#‘𝑠)) = (𝑥 − (#‘𝑆)))
14	13	adantr 480	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 − (#‘𝑠)) = (𝑥 − (#‘𝑆)))
15	12, 14	fveq12d 6109	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑡‘(𝑥 − (#‘𝑠))) = (𝑇‘(𝑥 − (#‘𝑆))))
16	9, 11, 15	ifbieq12d 4063	. . . 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 4663	. . 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 13156	. . 3 ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
19		ovex 6577	. . . 4 ⊢ (0..^((#‘𝑆) + (#‘𝑇))) ∈ V
20	19	mptex 6390	. . 3 ⊢ (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))) ∈ V
21	17, 18, 20	ovmpt2a 6689	. 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 493	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818   − cmin 10145  ..^cfzo 12334  #chash 12979   ++ cconcat 13148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-concat 13156

This theorem is referenced by:  ccatcl  13212  ccatlen  13213  ccatval1  13214  ccatval2  13215  ccatvalfn  13218  ccatalpha  13228  repswccat  13383  ccatco  13432  ofccat  13556  ccatmulgnn0dir  29945