MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgredeu Structured version   Visualization version   GIF version

Theorem efgredeu 18086
Description: There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
efgred.d 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
efgred.s 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
Assertion
Ref Expression
efgredeu (𝐴𝑊 → ∃!𝑑𝐷 𝑑 𝐴)
Distinct variable groups:   𝐴,𝑑   𝑦,𝑧   𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥   𝑚,𝑀   𝑥,𝑛,𝑀,𝑡,𝑣,𝑤   𝑘,𝑚,𝑡,𝑥,𝑇   𝑘,𝑑,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧,𝑊   ,𝑑,𝑚,𝑡,𝑥,𝑦,𝑧   𝑆,𝑑   𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑑,𝑚,𝑡
Allowed substitution hints:   𝐴(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,𝑑)   𝐼(𝑘,𝑑)   𝑀(𝑦,𝑧,𝑘,𝑑)

Proof of Theorem efgredeu
Dummy variables 𝑎 𝑏 𝑐 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . 5 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 efgval.r . . . . 5 = ( ~FG𝐼)
3 efgval2.m . . . . 5 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
4 efgval2.t . . . . 5 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
5 efgred.d . . . . 5 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
6 efgred.s . . . . 5 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
71, 2, 3, 4, 5, 6efgsfo 18073 . . . 4 𝑆:dom 𝑆onto𝑊
8 foelrn 6334 . . . 4 ((𝑆:dom 𝑆onto𝑊𝐴𝑊) → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆𝑎))
97, 8mpan 705 . . 3 (𝐴𝑊 → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆𝑎))
101, 2, 3, 4, 5, 6efgsdm 18064 . . . . . . . 8 (𝑎 ∈ dom 𝑆 ↔ (𝑎 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑎‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝑎))(𝑎𝑖) ∈ ran (𝑇‘(𝑎‘(𝑖 − 1)))))
1110simp2bi 1075 . . . . . . 7 (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∈ 𝐷)
1211adantl 482 . . . . . 6 ((𝐴𝑊𝑎 ∈ dom 𝑆) → (𝑎‘0) ∈ 𝐷)
131, 2, 3, 4, 5, 6efgsrel 18068 . . . . . . 7 (𝑎 ∈ dom 𝑆 → (𝑎‘0) (𝑆𝑎))
1413adantl 482 . . . . . 6 ((𝐴𝑊𝑎 ∈ dom 𝑆) → (𝑎‘0) (𝑆𝑎))
15 breq1 4616 . . . . . . 7 (𝑑 = (𝑎‘0) → (𝑑 (𝑆𝑎) ↔ (𝑎‘0) (𝑆𝑎)))
1615rspcev 3295 . . . . . 6 (((𝑎‘0) ∈ 𝐷 ∧ (𝑎‘0) (𝑆𝑎)) → ∃𝑑𝐷 𝑑 (𝑆𝑎))
1712, 14, 16syl2anc 692 . . . . 5 ((𝐴𝑊𝑎 ∈ dom 𝑆) → ∃𝑑𝐷 𝑑 (𝑆𝑎))
18 breq2 4617 . . . . . 6 (𝐴 = (𝑆𝑎) → (𝑑 𝐴𝑑 (𝑆𝑎)))
1918rexbidv 3045 . . . . 5 (𝐴 = (𝑆𝑎) → (∃𝑑𝐷 𝑑 𝐴 ↔ ∃𝑑𝐷 𝑑 (𝑆𝑎)))
2017, 19syl5ibrcom 237 . . . 4 ((𝐴𝑊𝑎 ∈ dom 𝑆) → (𝐴 = (𝑆𝑎) → ∃𝑑𝐷 𝑑 𝐴))
2120rexlimdva 3024 . . 3 (𝐴𝑊 → (∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆𝑎) → ∃𝑑𝐷 𝑑 𝐴))
229, 21mpd 15 . 2 (𝐴𝑊 → ∃𝑑𝐷 𝑑 𝐴)
231, 2efger 18052 . . . . . . 7 Er 𝑊
2423a1i 11 . . . . . 6 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → Er 𝑊)
25 simprl 793 . . . . . 6 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → 𝑑 𝐴)
26 simprr 795 . . . . . 6 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → 𝑐 𝐴)
2724, 25, 26ertr4d 7706 . . . . 5 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → 𝑑 𝑐)
281, 2, 3, 4, 5, 6efgrelex 18085 . . . . . 6 (𝑑 𝑐 → ∃𝑎 ∈ (𝑆 “ {𝑑})∃𝑏 ∈ (𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0))
29 fofn 6074 . . . . . . . . . . . . . 14 (𝑆:dom 𝑆onto𝑊𝑆 Fn dom 𝑆)
30 fniniseg 6294 . . . . . . . . . . . . . 14 (𝑆 Fn dom 𝑆 → (𝑎 ∈ (𝑆 “ {𝑑}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑑)))
317, 29, 30mp2b 10 . . . . . . . . . . . . 13 (𝑎 ∈ (𝑆 “ {𝑑}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆𝑎) = 𝑑))
3231simplbi 476 . . . . . . . . . . . 12 (𝑎 ∈ (𝑆 “ {𝑑}) → 𝑎 ∈ dom 𝑆)
3332ad2antrl 763 . . . . . . . . . . 11 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → 𝑎 ∈ dom 𝑆)
341, 2, 3, 4, 5, 6efgsval 18065 . . . . . . . . . . 11 (𝑎 ∈ dom 𝑆 → (𝑆𝑎) = (𝑎‘((#‘𝑎) − 1)))
3533, 34syl 17 . . . . . . . . . 10 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑆𝑎) = (𝑎‘((#‘𝑎) − 1)))
3631simprbi 480 . . . . . . . . . . 11 (𝑎 ∈ (𝑆 “ {𝑑}) → (𝑆𝑎) = 𝑑)
3736ad2antrl 763 . . . . . . . . . 10 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑆𝑎) = 𝑑)
38 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑑𝐷𝑐𝐷))
3938simpld 475 . . . . . . . . . . . . . . 15 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → 𝑑𝐷)
4037, 39eqeltrd 2698 . . . . . . . . . . . . . 14 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑆𝑎) ∈ 𝐷)
411, 2, 3, 4, 5, 6efgs1b 18070 . . . . . . . . . . . . . . 15 (𝑎 ∈ dom 𝑆 → ((𝑆𝑎) ∈ 𝐷 ↔ (#‘𝑎) = 1))
4233, 41syl 17 . . . . . . . . . . . . . 14 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((𝑆𝑎) ∈ 𝐷 ↔ (#‘𝑎) = 1))
4340, 42mpbid 222 . . . . . . . . . . . . 13 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (#‘𝑎) = 1)
4443oveq1d 6619 . . . . . . . . . . . 12 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((#‘𝑎) − 1) = (1 − 1))
45 1m1e0 11033 . . . . . . . . . . . 12 (1 − 1) = 0
4644, 45syl6eq 2671 . . . . . . . . . . 11 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((#‘𝑎) − 1) = 0)
4746fveq2d 6152 . . . . . . . . . 10 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑎‘((#‘𝑎) − 1)) = (𝑎‘0))
4835, 37, 473eqtr3rd 2664 . . . . . . . . 9 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑎‘0) = 𝑑)
49 fniniseg 6294 . . . . . . . . . . . . . 14 (𝑆 Fn dom 𝑆 → (𝑏 ∈ (𝑆 “ {𝑐}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑐)))
507, 29, 49mp2b 10 . . . . . . . . . . . . 13 (𝑏 ∈ (𝑆 “ {𝑐}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆𝑏) = 𝑐))
5150simplbi 476 . . . . . . . . . . . 12 (𝑏 ∈ (𝑆 “ {𝑐}) → 𝑏 ∈ dom 𝑆)
5251ad2antll 764 . . . . . . . . . . 11 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → 𝑏 ∈ dom 𝑆)
531, 2, 3, 4, 5, 6efgsval 18065 . . . . . . . . . . 11 (𝑏 ∈ dom 𝑆 → (𝑆𝑏) = (𝑏‘((#‘𝑏) − 1)))
5452, 53syl 17 . . . . . . . . . 10 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑆𝑏) = (𝑏‘((#‘𝑏) − 1)))
5550simprbi 480 . . . . . . . . . . 11 (𝑏 ∈ (𝑆 “ {𝑐}) → (𝑆𝑏) = 𝑐)
5655ad2antll 764 . . . . . . . . . 10 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑆𝑏) = 𝑐)
5738simprd 479 . . . . . . . . . . . . . . 15 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → 𝑐𝐷)
5856, 57eqeltrd 2698 . . . . . . . . . . . . . 14 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑆𝑏) ∈ 𝐷)
591, 2, 3, 4, 5, 6efgs1b 18070 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom 𝑆 → ((𝑆𝑏) ∈ 𝐷 ↔ (#‘𝑏) = 1))
6052, 59syl 17 . . . . . . . . . . . . . 14 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((𝑆𝑏) ∈ 𝐷 ↔ (#‘𝑏) = 1))
6158, 60mpbid 222 . . . . . . . . . . . . 13 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (#‘𝑏) = 1)
6261oveq1d 6619 . . . . . . . . . . . 12 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((#‘𝑏) − 1) = (1 − 1))
6362, 45syl6eq 2671 . . . . . . . . . . 11 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((#‘𝑏) − 1) = 0)
6463fveq2d 6152 . . . . . . . . . 10 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑏‘((#‘𝑏) − 1)) = (𝑏‘0))
6554, 56, 643eqtr3rd 2664 . . . . . . . . 9 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → (𝑏‘0) = 𝑐)
6648, 65eqeq12d 2636 . . . . . . . 8 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((𝑎‘0) = (𝑏‘0) ↔ 𝑑 = 𝑐))
6766biimpd 219 . . . . . . 7 ((((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) ∧ (𝑎 ∈ (𝑆 “ {𝑑}) ∧ 𝑏 ∈ (𝑆 “ {𝑐}))) → ((𝑎‘0) = (𝑏‘0) → 𝑑 = 𝑐))
6867rexlimdvva 3031 . . . . . 6 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → (∃𝑎 ∈ (𝑆 “ {𝑑})∃𝑏 ∈ (𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0) → 𝑑 = 𝑐))
6928, 68syl5 34 . . . . 5 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → (𝑑 𝑐𝑑 = 𝑐))
7027, 69mpd 15 . . . 4 (((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) ∧ (𝑑 𝐴𝑐 𝐴)) → 𝑑 = 𝑐)
7170ex 450 . . 3 ((𝐴𝑊 ∧ (𝑑𝐷𝑐𝐷)) → ((𝑑 𝐴𝑐 𝐴) → 𝑑 = 𝑐))
7271ralrimivva 2965 . 2 (𝐴𝑊 → ∀𝑑𝐷𝑐𝐷 ((𝑑 𝐴𝑐 𝐴) → 𝑑 = 𝑐))
73 breq1 4616 . . 3 (𝑑 = 𝑐 → (𝑑 𝐴𝑐 𝐴))
7473reu4 3382 . 2 (∃!𝑑𝐷 𝑑 𝐴 ↔ (∃𝑑𝐷 𝑑 𝐴 ∧ ∀𝑑𝐷𝑐𝐷 ((𝑑 𝐴𝑐 𝐴) → 𝑑 = 𝑐)))
7522, 72, 74sylanbrc 697 1 (𝐴𝑊 → ∃!𝑑𝐷 𝑑 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  {crab 2911  cdif 3552  c0 3891  {csn 4148  cop 4154  cotp 4156   ciun 4485   class class class wbr 4613  cmpt 4673   I cid 4984   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077   Fn wfn 5842  ontowfo 5845  cfv 5847  (class class class)co 6604  cmpt2 6606  1𝑜c1o 7498  2𝑜c2o 7499   Er wer 7684  0cc0 9880  1c1 9881  cmin 10210  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230   splice csplice 13235  ⟨“cs2 13523   ~FG cefg 18040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-s2 13530  df-efg 18043
This theorem is referenced by:  efgred2  18087  frgpnabllem2  18198
  Copyright terms: Public domain W3C validator