Theorem efgred 19530

Description: The reduced word that forms the base of the sequence in efgsval 19513 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 6918	. . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3978	. . . . . . 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 19511	. . . . . . . . . 10 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
10	9	fdmi 6680	. . . . . . . . . . 11 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
11	10	feq2i 6660	. . . . . . . . . 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 230	. . . . . . . . 9 ⊢ 𝑆:dom 𝑆⟶𝑊
13	12	ffvelcdmi 7034	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝑆 → (𝑆‘𝐴) ∈ 𝑊)
14	13	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (𝑆‘𝐴) ∈ 𝑊)
15	3, 14	sselid 3942	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (𝑆‘𝐴) ∈ Word (𝐼 × 2o))
16		lencl 14421	. . . . . 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 12453	. . . . 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 5109	. . . . . . 7 ⊢ (𝑐 = 0 → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < 0))
21	20	imbi1d 341	. . . . . 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 3212	. . . . 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 5109	. . . . . . 7 ⊢ (𝑐 = 𝑖 → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < 𝑖))
24	23	imbi1d 341	. . . . . 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 3212	. . . . 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 5109	. . . . . . 7 ⊢ (𝑐 = (𝑖 + 1) → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < (𝑖 + 1)))
27	26	imbi1d 341	. . . . . 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 3212	. . . . 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 5109	. . . . . . 7 ⊢ (𝑐 = ((♯‘(𝑆‘𝐴)) + 1) → ((♯‘(𝑆‘𝑎)) < 𝑐 ↔ (♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1)))
30	29	imbi1d 341	. . . . . 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 3212	. . . . 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 7034	. . . . . . . . . . 11 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) ∈ 𝑊)
33	3, 32	sselid 3942	. . . . . . . . . 10 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) ∈ Word (𝐼 × 2o))
34		lencl 14421	. . . . . . . . . 10 ⊢ ((𝑆‘𝑎) ∈ Word (𝐼 × 2o) → (♯‘(𝑆‘𝑎)) ∈ ℕ0)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑆 → (♯‘(𝑆‘𝑎)) ∈ ℕ0)
36		nn0nlt0 12439	. . . . . . . . 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 481	. . . . . 6 ⊢ ((𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) → ((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
40	39	rgen2 3194	. . . . 5 ⊢ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < 0 → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))
41		simpl1 1191	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . 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 5109	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(𝑆‘𝑐)) = 𝑖 → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝑐)) ↔ (♯‘(𝑆‘𝑎)) < 𝑖))
44	43	imbi1d 341	. . . . . . . . . . . . . . . 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 3212	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 1226	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . . 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 1229	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 19529	. . . . . . . . . . . 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 402	. . . . . . . . . . . 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 230	. . . . . . . . . . 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 1121	. . . . . . . . . 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 416	. . . . . . . . 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 3197	. . . . . . . 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 6847	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (♯‘(𝑆‘𝑐)) = (♯‘(𝑆‘𝑎)))
59	58	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → ((♯‘(𝑆‘𝑐)) = 𝑖 ↔ (♯‘(𝑆‘𝑎)) = 𝑖))
60		fveqeq2 6851	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → ((𝑆‘𝑐) = (𝑆‘𝑑) ↔ (𝑆‘𝑎) = (𝑆‘𝑑)))
61		fveq1 6841	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑎 → (𝑐‘0) = (𝑎‘0))
62	61	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → ((𝑐‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑑‘0)))
63	60, 62	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝑆‘𝑐) = (𝑆‘𝑑) → (𝑐‘0) = (𝑑‘0)) ↔ ((𝑆‘𝑎) = (𝑆‘𝑑) → (𝑎‘0) = (𝑑‘0))))
64	59, 63	imbi12d 344	. . . . . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (𝑆‘𝑑) = (𝑆‘𝑏))
66	65	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((𝑆‘𝑎) = (𝑆‘𝑑) ↔ (𝑆‘𝑎) = (𝑆‘𝑏)))
67		fveq1 6841	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (𝑑‘0) = (𝑏‘0))
68	67	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((𝑎‘0) = (𝑑‘0) ↔ (𝑎‘0) = (𝑏‘0)))
69	66, 68	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → (((𝑆‘𝑎) = (𝑆‘𝑑) → (𝑎‘0) = (𝑑‘0)) ↔ ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
70	69	imbi2d 340	. . . . . . . . 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 3227	. . . . . . . 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 217	. . . . . . 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 549	. . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆) → (♯‘(𝑆‘𝑎)) ∈ ℕ0)
75		nn0leltp1 12562	. . . . . . . . . . . . 13 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) ≤ 𝑖 ↔ (♯‘(𝑆‘𝑎)) < (𝑖 + 1)))
76		nn0re 12422	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝑆‘𝑎)) ∈ ℕ0 → (♯‘(𝑆‘𝑎)) ∈ ℝ)
77		nn0re 12422	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ)
78		leloe 11241	. . . . . . . . . . . . . 14 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((♯‘(𝑆‘𝑎)) ≤ 𝑖 ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
79	76, 77, 78	syl2an 596	. . . . . . . . . . . . 13 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) ≤ 𝑖 ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
80	75, 79	bitr3d 280	. . . . . . . . . . . 12 ⊢ (((♯‘(𝑆‘𝑎)) ∈ ℕ0 ∧ 𝑖 ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
81	80	ancoms 459	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ0 ∧ (♯‘(𝑆‘𝑎)) ∈ ℕ0) → ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
82	74, 81	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℕ0 ∧ (𝑎 ∈ dom 𝑆 ∧ 𝑏 ∈ dom 𝑆)) → ((♯‘(𝑆‘𝑎)) < (𝑖 + 1) ↔ ((♯‘(𝑆‘𝑎)) < 𝑖 ∨ (♯‘(𝑆‘𝑎)) = 𝑖)))
83	82	imbi1d 341	. . . . . . . . 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 960	. . . . . . . . 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 286	. . . . . . . 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 3210	. . . . . . 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 3135	. . . . . . 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 286	. . . . . 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	syl5ibr 245	. . . . 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 12598	. . . 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 12474	. . . 4 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (♯‘(𝑆‘𝐴)) ∈ ℝ)
93	92	ltp1d 12085	. . 3 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → (♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1))
94		2fveq3 6847	. . . . . 6 ⊢ (𝑎 = 𝐴 → (♯‘(𝑆‘𝑎)) = (♯‘(𝑆‘𝐴)))
95	94	breq1d 5115	. . . . 5 ⊢ (𝑎 = 𝐴 → ((♯‘(𝑆‘𝑎)) < ((♯‘(𝑆‘𝐴)) + 1) ↔ (♯‘(𝑆‘𝐴)) < ((♯‘(𝑆‘𝐴)) + 1)))
96		fveqeq2 6851	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ (𝑆‘𝐴) = (𝑆‘𝑏)))
97		fveq1 6841	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘0) = (𝐴‘0))
98	97	eqeq1d 2738	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎‘0) = (𝑏‘0) ↔ (𝐴‘0) = (𝑏‘0)))
99	96, 98	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)) ↔ ((𝑆‘𝐴) = (𝑆‘𝑏) → (𝐴‘0) = (𝑏‘0))))
100	95, 99	imbi12d 344	. . . 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 6842	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑆‘𝑏) = (𝑆‘𝐵))
102	101	eqeq2d 2747	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑆‘𝐴) = (𝑆‘𝑏) ↔ (𝑆‘𝐴) = (𝑆‘𝐵)))
103		fveq1 6841	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏‘0) = (𝐵‘0))
104	103	eqeq2d 2747	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴‘0) = (𝑏‘0) ↔ (𝐴‘0) = (𝐵‘0)))
105	102, 104	imbi12d 344	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑆‘𝐴) = (𝑆‘𝑏) → (𝐴‘0) = (𝑏‘0)) ↔ ((𝑆‘𝐴) = (𝑆‘𝐵) → (𝐴‘0) = (𝐵‘0))))
106	105	imbi2d 340	. . . 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 3590	. . 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 1117	1 ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆 ∧ (𝑆‘𝐴) = (𝑆‘𝐵)) → (𝐴‘0) = (𝐵‘0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407   ∖ cdif 3907  ∅c0 4282  {csn 4586  ⟨cop 4592  ⟨cotp 4594  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188   I cid 5530   × cxp 5631  dom cdm 5633  ran crn 5634  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1_oc1o 8405  2_oc2o 8406  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189   ≤ cle 11190   − cmin 11385  ℕ₀cn0 12413  ...cfz 13424  ..^cfzo 13567  ♯chash 14230  Word cword 14402   splice csplice 14637  ⟨“cs2 14730   ~_FG cefg 19488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-s2 14737

This theorem is referenced by:  efgrelexlemb  19532