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Theorem efgredlemg 19870

Description: Lemma for efgred 19876. (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 6959	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_o)) ⊆ Word (𝐼 × 2_o)
3	1, 2	eqsstri 3980	. . . . 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 19869	. . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))
17	16	simpld 500	. . . . 5 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
18	3, 17	sselid 3932	. . . 4 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_o))
19		lencl 14600	. . . 4 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
20	18, 19	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
21	20	nn0cnd 12594	. 2 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℂ)
22	16	simprd 501	. . . . 5 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
23	3, 22	sselid 3932	. . . 4 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_o))
24		lencl 14600	. . . 4 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
25	23, 24	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
26	25	nn0cnd 12594	. 2 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℂ)
27		2cnd 12346	. 2 ⊢ (𝜑 → 2 ∈ ℂ)
28	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	efgredlema 19868	. . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
29	28	simpld 500	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ)
30	1, 4, 5, 6, 7, 8	efgsdmi 19860	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((♯‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
31	10, 29, 30	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
32	14	fveq2i 6885	. . . . . . 7 ⊢ (𝐴‘𝐾) = (𝐴‘(((♯‘𝐴) − 1) − 1))
33	32	fveq2i 6885	. . . . . 6 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
34	33	rneqi 5925	. . . . 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 19854	. . . 4 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
37	17, 35, 36	syl2anc 596	. . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
38	28	simprd 501	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ)
39	1, 4, 5, 6, 7, 8	efgsdmi 19860	. . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((♯‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
40	11, 38, 39	syl2anc 596	. . . . . 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 6885	. . . . . . 7 ⊢ (𝐵‘𝐿) = (𝐵‘(((♯‘𝐵) − 1) − 1))
43	42	fveq2i 6885	. . . . . 6 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
44	43	rneqi 5925	. . . . 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 19854	. . . 4 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
47	22, 45, 46	syl2anc 596	. . 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 11443	1 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3078 {crab 3414 ∖ cdif 3899 ∅c0 4282 {csn 4587 ⟨cop 4593 ⟨cotp 4595 ∪ ciun 4954 class class class wbr 5107 ↦ cmpt 5190 I cid 5553 × cxp 5657 dom cdm 5659 ran crn 5660 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 1_oc1o 8451 2_oc2o 8452 0cc0 11127 1c1 11128 + caddc 11130 < clt 11270 − cmin 11468 ℕcn 12260 2c2 12322 ℕ₀cn0 12531 ...cfz 13563 ..^cfzo 13711 ♯chash 14396 Word cword 14580 splice csplice 14820 ⟨“cs2 14914 ~_FG cefg 19834

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3368 df-rab 3415 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 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13564 df-fzo 13712 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-substr 14711 df-pfx 14743 df-splice 14821 df-s2 14921

This theorem is used by: efgredleme 19871

W3C validator