Theorem efgred 19779

Description: The reduced word that forms the base of the sequence in efgsval 19762 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)

Hypotheses

Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))

Assertion

Ref	Expression
efgred	⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆 ∧ (𝑆‘𝐴) = (𝑆‘𝐵)) → (𝐴‘0) = (𝐵‘0))

Distinct variable groups:   𝑦,𝑧   𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥   𝑚,𝑀   𝑥,𝑛,𝑀,𝑡,𝑣,𝑤   𝑘,𝑚,𝑡,𝑥,𝑇   𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥   ∼ ,𝑚,𝑡,𝑥,𝑦,𝑧   𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑚,𝑡

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛)   ∼ (𝑤,𝑣,𝑘,𝑛)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝐼(𝑘)   𝑀(𝑦,𝑧,𝑘)

Proof of Theorem efgred

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		fviss 6939	. . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3980	. . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4		efgval.r	. . . . . . . . . . 11 ⊢ ∼ = ( ~FG ‘𝐼)
5		efgval2.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
6		efgval2.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
7		efgred.d	. . . . . . . . . . 11 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
8		efgred.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
9	1, 4, 5, 6, 7, 8	efgsf 19760	. . . . . . . . . 10 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
10	9	fdmi 6698	. . . . . . . . . . 11 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
11	10	feq2i 6678	. . . . . . . . . 10 ⊢ (𝑆:dom 𝑆⟶𝑊 ↔ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
12	9, 11	mpbir 233	. . . . . . . . 9 ⊢ 𝑆:dom 𝑆⟶𝑊
13	12	ffvelcdmi 7059	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑆 → (𝑆‘𝐴) ∈ 𝑊)
14	13	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (𝑆‘𝐴) ∈ 𝑊)
15	3, 14	sselid 3932	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (𝑆‘𝐴) ∈ Word (𝐼 × 2o))
16		lencl 14540	. . . . . 6 ⊢ ((𝑆‘𝐴) ∈ Word (𝐼 × 2o) → (♯‘(𝑆‘𝐴)) ∈ ℕ0)
17	15, 16	syl 17	. . . . 5 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (♯‘(𝑆‘𝐴)) ∈ ℕ0)
18		peano2nn0 12515	. . . . 5 ⊢ ((♯‘(𝑆‘𝐴)) ∈ ℕ0 → ((♯‘(𝑆‘𝐴)) + 1) ∈ ℕ0)
19	17, 18	syl 17	. . . 4 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((♯‘(𝑆‘𝐴)) + 1) ∈ ℕ0)
20		breq2 5101	. . . . . . 7 ⊢ (𝑐 = 0 → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < 0))
21	20	imbi1d 343	. . . . . 6 ⊢ (𝑐 = 0 → (((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
22	21	2ralbidv 3225	. . . . 5 ⊢ (𝑐 = 0 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
23		breq2 5101	. . . . . . 7 ⊢ (𝑐 = 𝑖 → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < 𝑖))
24	23	imbi1d 343	. . . . . 6 ⊢ (𝑐 = 𝑖 → (((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
25	24	2ralbidv 3225	. . . . 5 ⊢ (𝑐 = 𝑖 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
26		breq2 5101	. . . . . . 7 ⊢ (𝑐 = (𝑖 + 1) → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < (𝑖 + 1)))
27	26	imbi1d 343	. . . . . 6 ⊢ (𝑐 = (𝑖 + 1) → (((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
28	27	2ralbidv 3225	. . . . 5 ⊢ (𝑐 = (𝑖 + 1) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
29		breq2 5101	. . . . . . 7 ⊢ (𝑐 = ((♯‘(𝑆‘𝐴)) + 1) → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1)))
30	29	imbi1d 343	. . . . . 6 ⊢ (𝑐 = ((♯‘(𝑆‘𝐴)) + 1) → (((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
31	30	2ralbidv 3225	. . . . 5 ⊢ (𝑐 = ((♯‘(𝑆‘𝐴)) + 1) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑐 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
32	12	ffvelcdmi 7059	. . . . . . . . . . 11 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) ∈ 𝑊)
33	3, 32	sselid 3932	. . . . . . . . . 10 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) ∈ Word (𝐼 × 2o))
34		lencl 14540	. . . . . . . . . 10 ⊢ ((𝑆‘𝑎) ∈ Word (𝐼 × 2o) → (♯‘(𝑆‘𝑎)) ∈ ℕ0)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑆 → (♯‘(𝑆‘𝑎)) ∈ ℕ0)
36		nn0nlt0 12501	. . . . . . . . 9 ⊢ ((♯‘(𝑆‘𝑎)) ∈ ℕ0 → ¬ (♯‘(𝑆‘𝑎)) < 0)
37	35, 36	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ dom 𝑆 → ¬ (♯‘(𝑆‘𝑎)) < 0)
38	37	pm2.21d 121	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 → ((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
39	38	adantr 484	. . . . . 6 ⊢ ((𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) → ((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
40	39	rgen2 3201	. . . . 5 ⊢ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))
41		simpl1 1204	. . . . . . . . . . . . . 14 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
42		simpl3l 1241	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → (♯‘(𝑆‘𝑐)) = 𝑖)
43		breq2 5101	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(𝑆‘𝑐)) = 𝑖 → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝑐)) ↔ (♯‘(𝑆‘𝑎)) < 𝑖))
44	43	imbi1d 343	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(𝑆‘𝑐)) = 𝑖 → (((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝑐)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
45	44	2ralbidv 3225	. . . . . . . . . . . . . . 15 ⊢ ((♯‘(𝑆‘𝑐)) = 𝑖 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝑐)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
46	42, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝑐)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
47	41, 46	mpbird 259	. . . . . . . . . . . . 13 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝑐)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
48		simpl2l 1239	. . . . . . . . . . . . 13 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → 𝑐 ∈ dom 𝑆)
49		simpl2r 1240	. . . . . . . . . . . . 13 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → 𝑑 ∈ dom 𝑆)
50		simpl3r 1242	. . . . . . . . . . . . 13 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → (𝑆‘𝑐) = (𝑆‘𝑑))
51		simpr 488	. . . . . . . . . . . . 13 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)) → ¬ (𝑐‘0) = (𝑑‘0))
52	1, 4, 5, 6, 7, 8, 47, 48, 49, 50, 51	efgredlem 19778	. . . . . . . . . . . 12 ⊢ ¬ ((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0))
53		iman 405	. . . . . . . . . . . 12 ⊢ (((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) → (𝑐‘0) = (𝑑‘0)) ↔ ¬ ((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) ∧ ¬ (𝑐‘0) = (𝑑‘0)))
54	52, 53	mpbir 233	. . . . . . . . . . 11 ⊢ ((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆) ∧ ((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑))) → (𝑐‘0) = (𝑑‘0))
55	54	3expia 1133	. . . . . . . . . 10 ⊢ ((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆)) → (((♯‘(𝑆‘𝑐)) = 𝑖 ∧ (𝑆‘𝑐) = (𝑆‘𝑑)) → (𝑐‘0) = (𝑑‘0)))
56	55	expd 419	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ (𝑐 ∈ dom 𝑆 ∧ 𝑑 ∈ dom 𝑆)) → ((♯‘(𝑆‘𝑐)) = 𝑖 → ((𝑆‘𝑐) = (𝑆‘𝑑) → (𝑐‘0) = (𝑑‘0))))
57	56	ralrimivva 3204	. . . . . . . 8 ⊢ (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) → ∀𝑐 ∈ dom 𝑆∀𝑑 ∈ dom 𝑆((♯‘(𝑆‘𝑐)) = 𝑖 → ((𝑆‘𝑐) = (𝑆‘𝑑) → (𝑐‘0) = (𝑑‘0))))
58		2fveq3 6867	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (♯‘(𝑆‘𝑐)) = (♯‘(𝑆‘𝑎)))
59	58	eqeq1d 2763	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → ((♯‘(𝑆‘𝑐)) = 𝑖 ↔ (♯‘(𝑆‘𝑎)) = 𝑖))
60		fveqeq2 6871	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → ((𝑆‘𝑐) = (𝑆‘𝑑) ↔ (𝑆‘𝑎) = (𝑆‘𝑑)))
61		fveq1 6861	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
62	61	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
63	60, 62	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝑆‘𝑐) = (𝑆‘𝑑) → (𝑐‘0) = (𝑑‘0)) ↔ ((𝑆‘𝑎) = (𝑆‘𝑑) → (𝑎‘0) = (𝑑‘0))))
64	59, 63	imbi12d 346	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → (((♯‘(𝑆‘𝑐)) = 𝑖 → ((𝑆‘𝑐) = (𝑆‘𝑑) → (𝑐‘0) = (𝑑‘0))) ↔ ((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑑) → (𝑎‘0) = (𝑑‘0)))))
65		fveq2 6862	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (𝑆‘𝑑) = (𝑆‘𝑏))
66	65	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((𝑆‘𝑎) = (𝑆‘𝑑) ↔ (𝑆‘𝑎) = (𝑆‘𝑏)))
67		fveq1 6861	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
68	67	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
69	66, 68	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → (((𝑆‘𝑎) = (𝑆‘𝑑) → (𝑎‘0) = (𝑑‘0)) ↔ ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
70	69	imbi2d 342	. . . . . . . . 9 ⊢ (𝑑 = 𝑏 → (((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑑) → (𝑎‘0) = (𝑑‘0))) ↔ ((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
71	64, 70	cbvral2vw 3243	. . . . . . . 8 ⊢ (∀𝑐 ∈ dom 𝑆∀𝑑 ∈ dom 𝑆((♯‘(𝑆‘𝑐)) = 𝑖 → ((𝑆‘𝑐) = (𝑆‘𝑑) → (𝑐‘0) = (𝑑‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
72	57, 71	sylib 220	. . . . . . 7 ⊢ (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
73	72	ancli 556	. . . . . 6 ⊢ (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
74	35	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) → (♯‘(𝑆‘𝑎)) ∈ ℕ0)
75		nn0leltp1 12626	. . . . . . . . . . . . 13 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) ≤ 𝑖 ↔ (♯‘(𝑆‘𝑎)) < (𝑖 + 1)))
76		nn0re 12484	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝑆‘𝑎)) ∈ ℕ0 → (♯‘(𝑆‘𝑎)) ∈ ℝ)
77		nn0re 12484	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ)
78		leloe 11263	. . . . . . . . . . . . . 14 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((♯‘(𝑆‘𝑎)) ≤ 𝑖 ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
79	76, 77, 78	syl2an 605	. . . . . . . . . . . . 13 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) ≤ 𝑖 ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
80	75, 79	bitr3d 283	. . . . . . . . . . . 12 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
81	80	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ0 ∧ (♯‘(𝑆‘𝑎)) ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
82	74, 81	sylan2 602	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℕ0 ∧ (𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆)) → ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
83	82	imbi1d 343	. . . . . . . . 9 ⊢ ((𝑖 ∈ ℕ0 ∧ (𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆)) → (((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ (((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
84		jaob 974	. . . . . . . . 9 ⊢ ((((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ (((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
85	83, 84	bitrdi 289	. . . . . . . 8 ⊢ ((𝑖 ∈ ℕ0 ∧ (𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆)) → (((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ (((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))))
86	85	2ralbidva 3223	. . . . . . 7 ⊢ (𝑖 ∈ ℕ0 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆(((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))))
87		r19.26-2 3146	. . . . . . 7 ⊢ (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆(((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) ↔ (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
88	86, 87	bitrdi 289	. . . . . 6 ⊢ (𝑖 ∈ ℕ0 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ∧ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) = 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))))
89	73, 88	imbitrrid 248	. . . . 5 ⊢ (𝑖 ∈ ℕ0 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 𝑖 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (𝑖 + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))))
90	22, 25, 28, 31, 40, 89	nn0ind 12662	. . . 4 ⊢ (((♯‘(𝑆‘𝐴)) + 1) ∈ ℕ0 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
91	19, 90	syl 17	. . 3 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
92	17	nn0red 12537	. . . 4 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (♯‘(𝑆‘𝐴)) ∈ ℝ)
93	92	ltp1d 12116	. . 3 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1))
94		2fveq3 6867	. . . . . 6 ⊢ (𝑎 = 𝐴 → (♯‘(𝑆‘𝑎)) = (♯‘(𝑆‘𝐴)))
95	94	breq1d 5107	. . . . 5 ⊢ (𝑎 = 𝐴 → ((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) ↔ (♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1)))
96		fveqeq2 6871	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ (𝑆‘𝐴) = (𝑆‘𝑏)))
97		fveq1 6861	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘0) = (𝐴‘0))
98	97	eqeq1d 2763	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎‘0) = (𝑏‘0) ↔ (𝐴‘0) = (𝑏‘0)))
99	96, 98	imbi12d 346	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)) ↔ ((𝑆‘𝐴) = (𝑆‘𝑏) → (𝐴‘0) = (𝑏‘0))))
100	95, 99	imbi12d 346	. . . 4 ⊢ (𝑎 = 𝐴 → (((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝐴) = (𝑆‘𝑏) → (𝐴‘0) = (𝑏‘0)))))
101		fveq2 6862	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑆‘𝑏) = (𝑆‘𝐵))
102	101	eqeq2d 2772	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑆‘𝐴) = (𝑆‘𝑏) ↔ (𝑆‘𝐴) = (𝑆‘𝐵)))
103		fveq1 6861	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏‘0) = (𝐵‘0))
104	103	eqeq2d 2772	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴‘0) = (𝑏‘0) ↔ (𝐴‘0) = (𝐵‘0)))
105	102, 104	imbi12d 346	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑆‘𝐴) = (𝑆‘𝑏) → (𝐴‘0) = (𝑏‘0)) ↔ ((𝑆‘𝐴) = (𝑆‘𝐵) → (𝐴‘0) = (𝐵‘0))))
106	105	imbi2d 342	. . . 4 ⊢ (𝑏 = 𝐵 → (((♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝐴) = (𝑆‘𝑏) → (𝐴‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝐴) = (𝑆‘𝐵) → (𝐴‘0) = (𝐵‘0)))))
107	100, 106	rspc2v 3591	. . 3 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) → ((♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1) → ((𝑆‘𝐴) = (𝑆‘𝐵) → (𝐴‘0) = (𝐵‘0)))))
108	91, 93, 107	mp2d 49	. 2 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((𝑆‘𝐴) = (𝑆‘𝐵) → (𝐴‘0) = (𝐵‘0)))
109	108	3impia 1129	1 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆 ∧ (𝑆‘𝐴) = (𝑆‘𝐵)) → (𝐴‘0) = (𝐵‘0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ∖ cdif 3899  ∅c0 4283  {csn 4579  ⟨cop 4585  ⟨cotp 4587  ∪ ciun 4946   class class class wbr 5097   ↦ cmpt 5178   I cid 5537   × cxp 5641  dom cdm 5643  ran crn 5644  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1_oc1o 8424  2_oc2o 8425  ℝcr 11066  0cc0 11067  1c1 11068   + caddc 11070   < clt 11210   ≤ cle 11211   − cmin 11408  ℕ₀cn0 12475  ...cfz 13506  ..^cfzo 13653  ♯chash 14337  Word cword 14520   splice csplice 14756  ⟨“cs2 14848   ~_FG cefg 19737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-n0 12476  df-xnn0 12549  df-z 12563  df-uz 12834  df-rp 12988  df-fz 13507  df-fzo 13654  df-hash 14338  df-word 14521  df-concat 14578  df-s1 14604  df-substr 14649  df-pfx 14679  df-splice 14757  df-s2 14855

This theorem is referenced by:  efgrelexlemb  19781