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Theorem efgredlemg 19818

Description: Lemma for efgred 19824. (Contributed by Mario Carneiro, 4-Jun-2016.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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)))
efgredlem.1	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
efgredlem.2	⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
efgredlem.3	⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
efgredlem.4	⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
efgredlem.5	⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
efgredlemb.k	⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)
efgredlemb.l	⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)
efgredlemb.p	⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))
efgredlemb.q	⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))
efgredlemb.u	⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_o))
efgredlemb.v	⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o))
efgredlemb.6	⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
efgredlemb.7	⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))

**Assertion**
Ref	Expression
efgredlemg	⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿)))

Distinct variable groups: 𝑎,𝑏,𝐴 𝑦,𝑎,𝑧,𝑏 𝐿,𝑎,𝑏 𝐾,𝑎,𝑏 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑃 𝑚,𝑎,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑏 𝑈,𝑛,𝑣,𝑤,𝑦,𝑧 𝑘,𝑎,𝑇,𝑏,𝑚,𝑡,𝑥 𝑛,𝑉,𝑣,𝑤,𝑦,𝑧 𝑄,𝑛,𝑡,𝑣,𝑤,𝑦,𝑧 𝑊,𝑎,𝑏 𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥 ∼ ,𝑎,𝑏,𝑚,𝑡,𝑥,𝑦,𝑧 𝐵,𝑎,𝑏 𝑆,𝑎,𝑏 𝐼,𝑎,𝑏,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑎,𝑏,𝑚,𝑡 Allowed substitution hints: 𝜑(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑘, 𝑚, 𝑛, 𝑎, 𝑏) 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) 𝑃(𝑥, 𝑘, 𝑚, 𝑎, 𝑏) 𝑄(𝑥, 𝑘, 𝑚, 𝑎, 𝑏) ∼ (𝑤, 𝑣, 𝑘, 𝑛) 𝑆(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑘, 𝑚, 𝑛) 𝑇(𝑦, 𝑧, 𝑤, 𝑣, 𝑛) 𝑈(𝑥, 𝑡, 𝑘, 𝑚, 𝑎, 𝑏) 𝐼(𝑘) 𝐾(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑘, 𝑚, 𝑛) 𝐿(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑘, 𝑚, 𝑛) 𝑀(𝑦, 𝑧, 𝑘) 𝑉(𝑥, 𝑡, 𝑘, 𝑚, 𝑎, 𝑏)

**Proof of Theorem efgredlemg**
Step	Hyp	Ref	Expression
1		efgval.w	. . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
2		fviss 6958	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_o)) ⊆ Word (𝐼 × 2_o)
3	1, 2	eqsstri 3982	. . . . 5 ⊢ 𝑊 ⊆ Word (𝐼 × 2_o)
4		efgval.r	. . . . . . 7 ⊢ ∼ = ( ~_FG ‘𝐼)
5		efgval2.m	. . . . . . 7 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
6		efgval2.t	. . . . . . 7 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
7		efgred.d	. . . . . . 7 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
8		efgred.s	. . . . . . 7 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
9		efgredlem.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
10		efgredlem.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
11		efgredlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
12		efgredlem.4	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
13		efgredlem.5	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
14		efgredlemb.k	. . . . . . 7 ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)
15		efgredlemb.l	. . . . . . 7 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)
16	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	efgredlemf 19817	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))
17	16	simpld 499	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
18	3, 17	sselid 3934	. . . 4 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_o))
19		lencl 14577	. . . 4 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
20	18, 19	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
21	20	nn0cnd 12573	. 2 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℂ)
22	16	simprd 500	. . . . 5 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
23	3, 22	sselid 3934	. . . 4 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_o))
24		lencl 14577	. . . 4 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
25	23, 24	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
26	25	nn0cnd 12573	. 2 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℂ)
27		2cnd 12325	. 2 ⊢ (𝜑 → 2 ∈ ℂ)
28	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	efgredlema 19816	. . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
29	28	simpld 499	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ)
30	1, 4, 5, 6, 7, 8	efgsdmi 19808	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((♯‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
31	10, 29, 30	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
32	14	fveq2i 6884	. . . . . . 7 ⊢ (𝐴‘𝐾) = (𝐴‘(((♯‘𝐴) − 1) − 1))
33	32	fveq2i 6884	. . . . . 6 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
34	33	rneqi 5926	. . . . 5 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
35	31, 34	eleqtrrdi 2873	. . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾)))
36	1, 4, 5, 6	efgtlen 19802	. . . 4 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
37	17, 35, 36	syl2anc 595	. . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
38	28	simprd 500	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ)
39	1, 4, 5, 6, 7, 8	efgsdmi 19808	. . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((♯‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
40	11, 38, 39	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
41	12, 40	eqeltrd 2862	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
42	15	fveq2i 6884	. . . . . . 7 ⊢ (𝐵‘𝐿) = (𝐵‘(((♯‘𝐵) − 1) − 1))
43	42	fveq2i 6884	. . . . . 6 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
44	43	rneqi 5926	. . . . 5 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
45	41, 44	eleqtrrdi 2873	. . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿)))
46	1, 4, 5, 6	efgtlen 19802	. . . 4 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
47	22, 45, 46	syl2anc 595	. . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
48	37, 47	eqtr3d 2799	. 2 ⊢ (𝜑 → ((♯‘(𝐴‘𝐾)) + 2) = ((♯‘(𝐵‘𝐿)) + 2))
49	21, 26, 27, 48	addcan2ad 11422	1 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 {crab 3415 ∖ cdif 3901 ∅c0 4285 {csn 4588 ⟨cop 4594 ⟨cotp 4596 ∪ ciun 4955 class class class wbr 5108 ↦ cmpt 5191 I cid 5554 × cxp 5658 dom cdm 5660 ran crn 5661 ‘cfv 6536 (class class class)co 7412 ∈ cmpo 7414 1_oc1o 8444 2_oc2o 8445 0cc0 11106 1c1 11107 + caddc 11109 < clt 11249 − cmin 11447 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ...cfz 13541 ..^cfzo 13689 ♯chash 14373 Word cword 14557 splice csplice 14793 ⟨“cs2 14885 ~_FG cefg 19782

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-ot 4597 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-n0 12511 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-hash 14374 df-word 14558 df-concat 14615 df-s1 14641 df-substr 14686 df-pfx 14716 df-splice 14794 df-s2 14892

This theorem is used by: efgredleme 19819

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