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Theorem efgredlemg 19784

Description: Lemma for efgred 19790. (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 6946	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_o)) ⊆ Word (𝐼 × 2_o)
3	1, 2	eqsstri 3984	. . . . 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 19783	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))
17	16	simpld 498	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
18	3, 17	sselid 3936	. . . 4 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_o))
19		lencl 14548	. . . 4 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
21	20	nn0cnd 12546	. 2 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℂ)
22	16	simprd 499	. . . . 5 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
23	3, 22	sselid 3936	. . . 4 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_o))
24		lencl 14548	. . . 4 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
25	23, 24	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
26	25	nn0cnd 12546	. 2 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℂ)
27		2cnd 12298	. 2 ⊢ (𝜑 → 2 ∈ ℂ)
28	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	efgredlema 19782	. . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
29	28	simpld 498	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ)
30	1, 4, 5, 6, 7, 8	efgsdmi 19774	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((♯‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
31	10, 29, 30	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
32	14	fveq2i 6872	. . . . . . 7 ⊢ (𝐴‘𝐾) = (𝐴‘(((♯‘𝐴) − 1) − 1))
33	32	fveq2i 6872	. . . . . 6 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
34	33	rneqi 5915	. . . . 5 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
35	31, 34	eleqtrrdi 2875	. . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾)))
36	1, 4, 5, 6	efgtlen 19768	. . . 4 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
37	17, 35, 36	syl2anc 593	. . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
38	28	simprd 499	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ)
39	1, 4, 5, 6, 7, 8	efgsdmi 19774	. . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((♯‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
40	11, 38, 39	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
41	12, 40	eqeltrd 2864	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
42	15	fveq2i 6872	. . . . . . 7 ⊢ (𝐵‘𝐿) = (𝐵‘(((♯‘𝐵) − 1) − 1))
43	42	fveq2i 6872	. . . . . 6 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
44	43	rneqi 5915	. . . . 5 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
45	41, 44	eleqtrrdi 2875	. . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿)))
46	1, 4, 5, 6	efgtlen 19768	. . . 4 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
47	22, 45, 46	syl2anc 593	. . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
48	37, 47	eqtr3d 2801	. 2 ⊢ (𝜑 → ((♯‘(𝐴‘𝐾)) + 2) = ((♯‘(𝐵‘𝐿)) + 2))
49	21, 26, 27, 48	addcan2ad 11391	1 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ∀wral 3078 {crab 3416 ∖ cdif 3903 ∅c0 4287 {csn 4584 ⟨cop 4590 ⟨cotp 4592 ∪ ciun 4951 class class class wbr 5102 ↦ cmpt 5183 I cid 5543 × cxp 5647 dom cdm 5649 ran crn 5650 ‘cfv 6523 (class class class)co 7398 ∈ cmpo 7400 1_oc1o 8432 2_oc2o 8433 0cc0 11075 1c1 11076 + caddc 11078 < clt 11218 − cmin 11416 ℕcn 12212 2c2 12274 ℕ₀cn0 12483 ...cfz 13514 ..^cfzo 13661 ♯chash 14345 Word cword 14528 splice csplice 14764 ⟨“cs2 14856 ~_FG cefg 19748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-n0 12484 df-z 12571 df-uz 12842 df-fz 13515 df-fzo 13662 df-hash 14346 df-word 14529 df-concat 14586 df-s1 14612 df-substr 14657 df-pfx 14687 df-splice 14765 df-s2 14863

This theorem is referenced by: efgredleme 19785

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