ccatf1 - Mathbox for Thierry Arnoux

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Theorem ccatf1 32843

Description: Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)

**Hypotheses**
Ref	Expression
ccatf1.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
ccatf1.a	⊢ (𝜑 → 𝐴 ∈ Word 𝑆)
ccatf1.b	⊢ (𝜑 → 𝐵 ∈ Word 𝑆)
ccatf1.1	⊢ (𝜑 → 𝐴:dom 𝐴–1-1→𝑆)
ccatf1.2	⊢ (𝜑 → 𝐵:dom 𝐵–1-1→𝑆)
ccatf1.3	⊢ (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)

**Assertion**
Ref	Expression
ccatf1	⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆)

Proof of Theorem ccatf1 Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatf1.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Word 𝑆)
2		ccatf1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Word 𝑆)
3		ccatcl 14515	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
4	1, 2, 3	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴 ++ 𝐵) ∈ Word 𝑆)
5		wrdf 14459	. . . 4 ⊢ ((𝐴 ++ 𝐵) ∈ Word 𝑆 → (𝐴 ++ 𝐵):(0..^(♯‘(𝐴 ++ 𝐵)))⟶𝑆)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ++ 𝐵):(0..^(♯‘(𝐴 ++ 𝐵)))⟶𝑆)
7	6	ffdmd 6700	. 2 ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)⟶𝑆)
8		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗))
9		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(♯‘𝐴)) → 𝑖 ∈ (0..^(♯‘𝐴)))
10		ccatval1 14518	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
11	1, 2, 9, 10	syl2an3an 1424	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
12	11	ad4ant13 751	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
13		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑗 ∈ (0..^(♯‘𝐴)))
14		ccatval1 14518	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
15	1, 2, 13, 14	syl2an3an 1424	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
16	15	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
17	8, 12, 16	3eqtr3d 2772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑖) = (𝐴‘𝑗))
18		wrddm 14462	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Word 𝑆 → dom 𝐴 = (0..^(♯‘𝐴)))
19	1, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐴 = (0..^(♯‘𝐴)))
20		ccatf1.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:dom 𝐴–1-1→𝑆)
21		f1eq2 6734	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐴 = (0..^(♯‘𝐴)) → (𝐴:dom 𝐴–1-1→𝑆 ↔ 𝐴:(0..^(♯‘𝐴))–1-1→𝑆))
22	21	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝐴 = (0..^(♯‘𝐴)) ∧ 𝐴:dom 𝐴–1-1→𝑆) → 𝐴:(0..^(♯‘𝐴))–1-1→𝑆)
23	19, 20, 22	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))–1-1→𝑆)
24		dff13 7211	. . . . . . . . . . . . . . . 16 ⊢ (𝐴:(0..^(♯‘𝐴))–1-1→𝑆 ↔ (𝐴:(0..^(♯‘𝐴))⟶𝑆 ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))∀𝑗 ∈ (0..^(♯‘𝐴))((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗)))
25	24	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝐴:(0..^(♯‘𝐴))–1-1→𝑆 → ∀𝑖 ∈ (0..^(♯‘𝐴))∀𝑗 ∈ (0..^(♯‘𝐴))((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
26	23, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝐴))∀𝑗 ∈ (0..^(♯‘𝐴))((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
27	26	r19.21bi 3227	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → ∀𝑗 ∈ (0..^(♯‘𝐴))((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
28	27	r19.21bi 3227	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
29	28	adantllr 719	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
30	17, 29	mpd 15	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑖 = 𝑗)
31	30	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
32	31	adantllr 719	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
33		f1fun 6740	. . . . . . . . . . . . . . . 16 ⊢ (𝐴:dom 𝐴–1-1→𝑆 → Fun 𝐴)
34	20, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐴)
35		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → 𝑖 ∈ (0..^(♯‘𝐴)))
36	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → dom 𝐴 = (0..^(♯‘𝐴)))
37	35, 36	eleqtrrd 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → 𝑖 ∈ dom 𝐴)
38		fvelrn 7030	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐴 ∧ 𝑖 ∈ dom 𝐴) → (𝐴‘𝑖) ∈ ran 𝐴)
39	34, 37, 38	syl2an2r 685	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑖) ∈ ran 𝐴)
40	39	ad4ant13 751	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ ran 𝐴)
41		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗))
42	11	ad4ant13 751	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
43	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐴 ∈ Word 𝑆)
44	2	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐵 ∈ Word 𝑆)
45		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))))
46		ccatlen 14516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
47	1, 2, 46	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
48	47	oveq2d 7385	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) = ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
49	48	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) = ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
50	45, 49	eleqtrd 2830	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑗 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
51		ccatval2 14519	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑗 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐵‘(𝑗 − (♯‘𝐴))))
52	43, 44, 50, 51	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐵‘(𝑗 − (♯‘𝐴))))
53	52	ad4ant14 752	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐵‘(𝑗 − (♯‘𝐴))))
54	41, 42, 53	3eqtr3d 2772	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) = (𝐵‘(𝑗 − (♯‘𝐴))))
55		ccatf1.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵:dom 𝐵–1-1→𝑆)
56		f1fun 6740	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵:dom 𝐵–1-1→𝑆 → Fun 𝐵)
57	55, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Fun 𝐵)
58		lencl 14474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ Word 𝑆 → (♯‘𝐵) ∈ ℕ₀)
59	2, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
60	59	nn0zd 12531	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
61	60	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (♯‘𝐵) ∈ ℤ)
62		fzosubel3 13663	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ∧ (♯‘𝐵) ∈ ℤ) → (𝑗 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
63	50, 61, 62	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
64		wrddm 14462	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Word 𝑆 → dom 𝐵 = (0..^(♯‘𝐵)))
65	2, 64	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐵 = (0..^(♯‘𝐵)))
66	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → dom 𝐵 = (0..^(♯‘𝐵)))
67	63, 66	eleqtrrd 2831	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 − (♯‘𝐴)) ∈ dom 𝐵)
68		fvelrn 7030	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐵 ∧ (𝑗 − (♯‘𝐴)) ∈ dom 𝐵) → (𝐵‘(𝑗 − (♯‘𝐴))) ∈ ran 𝐵)
69	57, 67, 68	syl2an2r 685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑗 − (♯‘𝐴))) ∈ ran 𝐵)
70	69	ad4ant14 752	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑗 − (♯‘𝐴))) ∈ ran 𝐵)
71	54, 70	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ ran 𝐵)
72	40, 71	elind 4159	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ (ran 𝐴 ∩ ran 𝐵))
73		ccatf1.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)
74	73	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (ran 𝐴 ∩ ran 𝐵) = ∅)
75	72, 74	eleqtrd 2830	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ ∅)
76		noel 4297	. . . . . . . . . . . 12 ⊢ ¬ (𝐴‘𝑖) ∈ ∅
77	76	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ¬ (𝐴‘𝑖) ∈ ∅)
78	75, 77	pm2.21dd 195	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 = 𝑗)
79	78	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
80	79	adantllr 719	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
81		wrddm 14462	. . . . . . . . . . . . 13 ⊢ ((𝐴 ++ 𝐵) ∈ Word 𝑆 → dom (𝐴 ++ 𝐵) = (0..^(♯‘(𝐴 ++ 𝐵))))
82	4, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝐴 ++ 𝐵) = (0..^(♯‘(𝐴 ++ 𝐵))))
83	82	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ dom (𝐴 ++ 𝐵) ↔ 𝑗 ∈ (0..^(♯‘(𝐴 ++ 𝐵)))))
84	83	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) → 𝑗 ∈ (0..^(♯‘(𝐴 ++ 𝐵))))
85		lencl 14474	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑆 → (♯‘𝐴) ∈ ℕ₀)
86	1, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
87	86	nn0zd 12531	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
88	87	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) → (♯‘𝐴) ∈ ℤ)
89		fzospliti 13628	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^(♯‘(𝐴 ++ 𝐵))) ∧ (♯‘𝐴) ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐴)) ∨ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
90	84, 88, 89	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) → (𝑗 ∈ (0..^(♯‘𝐴)) ∨ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
91	90	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ (0..^(♯‘𝐴)) ∨ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
92	32, 80, 91	mpjaod 860	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → 𝑖 = 𝑗)
93	92	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
94	93	adantlrl 720	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
95		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑗 ∈ (0..^(♯‘𝐴)))
96	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → dom 𝐴 = (0..^(♯‘𝐴)))
97	95, 96	eleqtrrd 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑗 ∈ dom 𝐴)
98		fvelrn 7030	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐴 ∧ 𝑗 ∈ dom 𝐴) → (𝐴‘𝑗) ∈ ran 𝐴)
99	34, 97, 98	syl2an2r 685	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ran 𝐴)
100	99	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ran 𝐴)
101		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗))
102	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐴 ∈ Word 𝑆)
103	2	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐵 ∈ Word 𝑆)
104		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))))
105	48	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) = ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
106	104, 105	eleqtrd 2830	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
107		ccatval2 14519	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑖 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
108	102, 103, 106, 107	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
109	108	ad4ant13 751	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
110	15	ad4ant14 752	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
111	101, 109, 110	3eqtr3rd 2773	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) = (𝐵‘(𝑖 − (♯‘𝐴))))
112	60	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (♯‘𝐵) ∈ ℤ)
113		fzosubel3 13663	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ∧ (♯‘𝐵) ∈ ℤ) → (𝑖 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
114	106, 112, 113	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
115	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → dom 𝐵 = (0..^(♯‘𝐵)))
116	114, 115	eleqtrrd 2831	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) ∈ dom 𝐵)
117		fvelrn 7030	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐵 ∧ (𝑖 − (♯‘𝐴)) ∈ dom 𝐵) → (𝐵‘(𝑖 − (♯‘𝐴))) ∈ ran 𝐵)
118	57, 116, 117	syl2an2r 685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑖 − (♯‘𝐴))) ∈ ran 𝐵)
119	118	ad4ant13 751	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐵‘(𝑖 − (♯‘𝐴))) ∈ ran 𝐵)
120	111, 119	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ran 𝐵)
121	100, 120	elind 4159	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ (ran 𝐴 ∩ ran 𝐵))
122	73	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (ran 𝐴 ∩ ran 𝐵) = ∅)
123	121, 122	eleqtrd 2830	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ∅)
124		noel 4297	. . . . . . . . . . . 12 ⊢ ¬ (𝐴‘𝑗) ∈ ∅
125	124	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ¬ (𝐴‘𝑗) ∈ ∅)
126	123, 125	pm2.21dd 195	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑖 = 𝑗)
127	126	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
128	127	adantllr 719	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
129		elfzoelz 13596	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 ∈ ℤ)
130	129	zcnd 12615	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 ∈ ℂ)
131	130	ad2antlr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 ∈ ℂ)
132		elfzoelz 13596	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑗 ∈ ℤ)
133	132	zcnd 12615	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑗 ∈ ℂ)
134	133	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑗 ∈ ℂ)
135	86	nn0cnd 12481	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ)
136	135	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (♯‘𝐴) ∈ ℂ)
137	55	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐵:dom 𝐵–1-1→𝑆)
138	116	ad4ant13 751	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) ∈ dom 𝐵)
139	67	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 − (♯‘𝐴)) ∈ dom 𝐵)
140	138, 139	jca 511	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝑖 − (♯‘𝐴)) ∈ dom 𝐵 ∧ (𝑗 − (♯‘𝐴)) ∈ dom 𝐵))
141		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗))
142	108	ad4ant13 751	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
143	52	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐵‘(𝑗 − (♯‘𝐴))))
144	141, 142, 143	3eqtr3d 2772	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑖 − (♯‘𝐴))) = (𝐵‘(𝑗 − (♯‘𝐴))))
145		f1veqaeq 7213	. . . . . . . . . . . . 13 ⊢ ((𝐵:dom 𝐵–1-1→𝑆 ∧ ((𝑖 − (♯‘𝐴)) ∈ dom 𝐵 ∧ (𝑗 − (♯‘𝐴)) ∈ dom 𝐵)) → ((𝐵‘(𝑖 − (♯‘𝐴))) = (𝐵‘(𝑗 − (♯‘𝐴))) → (𝑖 − (♯‘𝐴)) = (𝑗 − (♯‘𝐴))))
146	145	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐵:dom 𝐵–1-1→𝑆 ∧ ((𝑖 − (♯‘𝐴)) ∈ dom 𝐵 ∧ (𝑗 − (♯‘𝐴)) ∈ dom 𝐵)) ∧ (𝐵‘(𝑖 − (♯‘𝐴))) = (𝐵‘(𝑗 − (♯‘𝐴)))) → (𝑖 − (♯‘𝐴)) = (𝑗 − (♯‘𝐴)))
147	137, 140, 144, 146	syl21anc 837	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) = (𝑗 − (♯‘𝐴)))
148	131, 134, 136, 147	subcan2d 11551	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 = 𝑗)
149	148	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
150	149	adantllr 719	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
151	90	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ (0..^(♯‘𝐴)) ∨ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
152	128, 150, 151	mpjaod 860	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 = 𝑗)
153	152	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
154	153	adantlrl 720	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
155	82	eleq2d 2814	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ dom (𝐴 ++ 𝐵) ↔ 𝑖 ∈ (0..^(♯‘(𝐴 ++ 𝐵)))))
156	155	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (𝐴 ++ 𝐵)) → 𝑖 ∈ (0..^(♯‘(𝐴 ++ 𝐵))))
157	87	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (𝐴 ++ 𝐵)) → (♯‘𝐴) ∈ ℤ)
158		fzospliti 13628	. . . . . . . 8 ⊢ ((𝑖 ∈ (0..^(♯‘(𝐴 ++ 𝐵))) ∧ (♯‘𝐴) ∈ ℤ) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
159	156, 157, 158	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (𝐴 ++ 𝐵)) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
160	159	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
161	160	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
162	94, 154, 161	mpjaod 860	. . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → 𝑖 = 𝑗)
163	162	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) → (((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗) → 𝑖 = 𝑗))
164	163	ralrimivva 3178	. 2 ⊢ (𝜑 → ∀𝑖 ∈ dom (𝐴 ++ 𝐵)∀𝑗 ∈ dom (𝐴 ++ 𝐵)(((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗) → 𝑖 = 𝑗))
165		dff13 7211	. 2 ⊢ ((𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆 ↔ ((𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)⟶𝑆 ∧ ∀𝑖 ∈ dom (𝐴 ++ 𝐵)∀𝑗 ∈ dom (𝐴 ++ 𝐵)(((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗) → 𝑖 = 𝑗)))
166	7, 164, 165	sylanbrc 583	1 ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∩ cin 3910 ∅c0 4292 dom cdm 5631 ran crn 5632 Fun wfun 6493 ⟶wf 6495 –1-1→wf1 6496 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 0cc0 11044 + caddc 11047 − cmin 11381 ℕ₀cn0 12418 ℤcz 12505 ..^cfzo 13591 ♯chash 14271 Word cword 14454 ++ cconcat 14511

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-hash 14272 df-word 14455 df-concat 14512

This theorem is referenced by: cycpmco2f1 33054

W3C validator