ccatf1 - Mathbox for Thierry Arnoux

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Theorem ccatf1 30625

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 13926	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
4	1, 2, 3	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐴 ++ 𝐵) ∈ Word 𝑆)
5		wrdf 13867	. . . 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 6537	. 2 ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)⟶𝑆)
8		simpllr 774	. . . . . . . . . . . 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 13930	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
11	1, 2, 9, 10	syl2an3an 1418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
12	11	ad4ant13 749	. . . . . . . . . . . 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 13930	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
15	1, 2, 13, 14	syl2an3an 1418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
16	15	ad4ant14 750	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐴‘𝑗))
17	8, 12, 16	3eqtr3d 2864	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑖) = (𝐴‘𝑗))
18		wrddm 13869	. . . . . . . . . . . . . . . . 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 6571	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐴 = (0..^(♯‘𝐴)) → (𝐴:dom 𝐴–1-1→𝑆 ↔ 𝐴:(0..^(♯‘𝐴))–1-1→𝑆))
22	21	biimpa 479	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝐴 = (0..^(♯‘𝐴)) ∧ 𝐴:dom 𝐴–1-1→𝑆) → 𝐴:(0..^(♯‘𝐴))–1-1→𝑆)
23	19, 20, 22	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))–1-1→𝑆)
24		dff13 7013	. . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . 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 3208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → ∀𝑗 ∈ (0..^(♯‘𝐴))((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
28	27	r19.21bi 3208	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴‘𝑖) = (𝐴‘𝑗) → 𝑖 = 𝑗))
29	28	adantllr 717	. . . . . . . . . . 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 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
32	31	adantllr 717	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
33		f1fun 6577	. . . . . . . . . . . . . . . 16 ⊢ (𝐴:dom 𝐴–1-1→𝑆 → Fun 𝐴)
34	20, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐴)
35		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → 𝑖 ∈ (0..^(♯‘𝐴)))
36	19	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → dom 𝐴 = (0..^(♯‘𝐴)))
37	35, 36	eleqtrrd 2916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → 𝑖 ∈ dom 𝐴)
38		fvelrn 6844	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐴 ∧ 𝑖 ∈ dom 𝐴) → (𝐴‘𝑖) ∈ ran 𝐴)
39	34, 37, 38	syl2an2r 683	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑖) ∈ ran 𝐴)
40	39	ad4ant13 749	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ ran 𝐴)
41		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗))
42	11	ad4ant13 749	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐴‘𝑖))
43	1	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐴 ∈ Word 𝑆)
44	2	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐵 ∈ Word 𝑆)
45		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))))
46		ccatlen 13927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
47	1, 2, 46	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
48	47	oveq2d 7172	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) = ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
49	48	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) = ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
50	45, 49	eleqtrd 2915	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑗 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
51		ccatval2 13932	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑗 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐵‘(𝑗 − (♯‘𝐴))))
52	43, 44, 50, 51	syl3anc 1367	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑗) = (𝐵‘(𝑗 − (♯‘𝐴))))
53	52	ad4ant14 750	. . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . 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 6577	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵:dom 𝐵–1-1→𝑆 → Fun 𝐵)
57	55, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Fun 𝐵)
58		lencl 13883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ Word 𝑆 → (♯‘𝐵) ∈ ℕ₀)
59	2, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
60	59	nn0zd 12086	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
61	60	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (♯‘𝐵) ∈ ℤ)
62		fzosubel3 13099	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ∧ (♯‘𝐵) ∈ ℤ) → (𝑗 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
63	50, 61, 62	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
64		wrddm 13869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Word 𝑆 → dom 𝐵 = (0..^(♯‘𝐵)))
65	2, 64	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐵 = (0..^(♯‘𝐵)))
66	65	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → dom 𝐵 = (0..^(♯‘𝐵)))
67	63, 66	eleqtrrd 2916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 − (♯‘𝐴)) ∈ dom 𝐵)
68		fvelrn 6844	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐵 ∧ (𝑗 − (♯‘𝐴)) ∈ dom 𝐵) → (𝐵‘(𝑗 − (♯‘𝐴))) ∈ ran 𝐵)
69	57, 67, 68	syl2an2r 683	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑗 − (♯‘𝐴))) ∈ ran 𝐵)
70	69	ad4ant14 750	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑗 − (♯‘𝐴))) ∈ ran 𝐵)
71	54, 70	eqeltrd 2913	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ ran 𝐵)
72	40, 71	elind 4171	. . . . . . . . . . . 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 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (ran 𝐴 ∩ ran 𝐵) = ∅)
75	72, 74	eleqtrd 2915	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐴‘𝑖) ∈ ∅)
76		noel 4296	. . . . . . . . . . . 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 197	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 = 𝑗)
79	78	ex 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
80	79	adantllr 717	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
81		wrddm 13869	. . . . . . . . . . . . 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 2898	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ dom (𝐴 ++ 𝐵) ↔ 𝑗 ∈ (0..^(♯‘(𝐴 ++ 𝐵)))))
84	83	biimpa 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) → 𝑗 ∈ (0..^(♯‘(𝐴 ++ 𝐵))))
85		lencl 13883	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑆 → (♯‘𝐴) ∈ ℕ₀)
86	1, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
87	86	nn0zd 12086	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
88	87	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) → (♯‘𝐴) ∈ ℤ)
89		fzospliti 13070	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^(♯‘(𝐴 ++ 𝐵))) ∧ (♯‘𝐴) ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐴)) ∨ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
90	84, 88, 89	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) → (𝑗 ∈ (0..^(♯‘𝐴)) ∨ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
91	90	ad2antrr 724	. . . . . . . 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 856	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ (0..^(♯‘𝐴))) → 𝑖 = 𝑗)
93	92	ex 415	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
94	93	adantlrl 718	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
95		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑗 ∈ (0..^(♯‘𝐴)))
96	19	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → dom 𝐴 = (0..^(♯‘𝐴)))
97	95, 96	eleqtrrd 2916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑗 ∈ dom 𝐴)
98		fvelrn 6844	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐴 ∧ 𝑗 ∈ dom 𝐴) → (𝐴‘𝑗) ∈ ran 𝐴)
99	34, 97, 98	syl2an2r 683	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ran 𝐴)
100	99	ad4ant14 750	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ran 𝐴)
101		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗))
102	1	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐴 ∈ Word 𝑆)
103	2	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝐵 ∈ Word 𝑆)
104		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))))
105	48	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) = ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
106	104, 105	eleqtrd 2915	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
107		ccatval2 13932	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆 ∧ 𝑖 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
108	102, 103, 106, 107	syl3anc 1367	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
109	108	ad4ant13 749	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝑖) = (𝐵‘(𝑖 − (♯‘𝐴))))
110	15	ad4ant14 750	. . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) = (𝐵‘(𝑖 − (♯‘𝐴))))
112	60	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (♯‘𝐵) ∈ ℤ)
113		fzosubel3 13099	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ∧ (♯‘𝐵) ∈ ℤ) → (𝑖 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
114	106, 112, 113	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) ∈ (0..^(♯‘𝐵)))
115	65	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → dom 𝐵 = (0..^(♯‘𝐵)))
116	114, 115	eleqtrrd 2916	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) ∈ dom 𝐵)
117		fvelrn 6844	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐵 ∧ (𝑖 − (♯‘𝐴)) ∈ dom 𝐵) → (𝐵‘(𝑖 − (♯‘𝐴))) ∈ ran 𝐵)
118	57, 116, 117	syl2an2r 683	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑖 − (♯‘𝐴))) ∈ ran 𝐵)
119	118	ad4ant13 749	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐵‘(𝑖 − (♯‘𝐴))) ∈ ran 𝐵)
120	111, 119	eqeltrd 2913	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ran 𝐵)
121	100, 120	elind 4171	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ (ran 𝐴 ∩ ran 𝐵))
122	73	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (ran 𝐴 ∩ ran 𝐵) = ∅)
123	121, 122	eleqtrd 2915	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑗) ∈ ∅)
124		noel 4296	. . . . . . . . . . . 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 197	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ (0..^(♯‘𝐴))) → 𝑖 = 𝑗)
127	126	ex 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
128	127	adantllr 717	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ (0..^(♯‘𝐴)) → 𝑖 = 𝑗))
129		elfzoelz 13039	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 ∈ ℤ)
130	129	zcnd 12089	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 ∈ ℂ)
131	130	ad2antlr 725	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 ∈ ℂ)
132		elfzoelz 13039	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑗 ∈ ℤ)
133	132	zcnd 12089	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑗 ∈ ℂ)
134	133	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑗 ∈ ℂ)
135	86	nn0cnd 11958	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ)
136	135	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (♯‘𝐴) ∈ ℂ)
137	55	ad3antrrr 728	. . . . . . . . . . . 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 749	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑖 − (♯‘𝐴)) ∈ dom 𝐵)
139	67	ad4ant14 750	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 − (♯‘𝐴)) ∈ dom 𝐵)
140	138, 139	jca 514	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 749	. . . . . . . . . . . . 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 750	. . . . . . . . . . . . 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 2864	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝐵‘(𝑖 − (♯‘𝐴))) = (𝐵‘(𝑗 − (♯‘𝐴))))
145		f1veqaeq 7015	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 835	. . . . . . . . . . 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 11039	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 = 𝑗)
149	148	ex 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → (𝑗 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
150	149	adantllr 717	. . . . . . . 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 724	. . . . . . . 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 856	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) ∧ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))) → 𝑖 = 𝑗)
153	152	ex 415	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵)) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → (𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵))) → 𝑖 = 𝑗))
154	153	adantlrl 718	. . . . 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 2898	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ dom (𝐴 ++ 𝐵) ↔ 𝑖 ∈ (0..^(♯‘(𝐴 ++ 𝐵)))))
156	155	biimpa 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (𝐴 ++ 𝐵)) → 𝑖 ∈ (0..^(♯‘(𝐴 ++ 𝐵))))
157	87	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (𝐴 ++ 𝐵)) → (♯‘𝐴) ∈ ℤ)
158		fzospliti 13070	. . . . . . . 8 ⊢ ((𝑖 ∈ (0..^(♯‘(𝐴 ++ 𝐵))) ∧ (♯‘𝐴) ∈ ℤ) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
159	156, 157, 158	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (𝐴 ++ 𝐵)) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
160	159	adantrr 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) → (𝑖 ∈ (0..^(♯‘𝐴)) ∨ 𝑖 ∈ ((♯‘𝐴)..^(♯‘(𝐴 ++ 𝐵)))))
161	160	adantr 483	. . . . 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 856	. . . 4 ⊢ (((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) ∧ ((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗)) → 𝑖 = 𝑗)
163	162	ex 415	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ dom (𝐴 ++ 𝐵) ∧ 𝑗 ∈ dom (𝐴 ++ 𝐵))) → (((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗) → 𝑖 = 𝑗))
164	163	ralrimivva 3191	. 2 ⊢ (𝜑 → ∀𝑖 ∈ dom (𝐴 ++ 𝐵)∀𝑗 ∈ dom (𝐴 ++ 𝐵)(((𝐴 ++ 𝐵)‘𝑖) = ((𝐴 ++ 𝐵)‘𝑗) → 𝑖 = 𝑗))
165		dff13 7013	. 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 585	1 ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 ∅c0 4291 dom cdm 5555 ran crn 5556 Fun wfun 6349 ⟶wf 6351 –1-1→wf1 6352 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 + caddc 10540 − cmin 10870 ℕ₀cn0 11898 ℤcz 11982 ..^cfzo 13034 ♯chash 13691 Word cword 13862 ++ cconcat 13922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923

This theorem is referenced by: cycpmco2f1 30766

W3C validator