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Mirrors > Home > MPE Home > Th. List > efgredlemb | Structured version Visualization version GIF version |
Description: The reduced word that forms the base of the sequence in efgsval 19647 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.) |
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))) |
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 × 2o)) |
efgredlemb.v | ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) |
efgredlemb.6 | ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) |
efgredlemb.7 | ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) |
efgredlemb.8 | ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) |
Ref | Expression |
---|---|
efgredlemb | ⊢ ¬ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
2 | efgval.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
3 | efgval2.m | . . . . 5 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
4 | efgval2.t | . . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
5 | efgred.d | . . . . 5 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
6 | efgred.s | . . . . 5 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | |
7 | efgredlem.1 | . . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) | |
8 | efgredlem.4 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) | |
9 | fveq2 6891 | . . . . . . . . . 10 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (♯‘(𝑆‘𝐴)) = (♯‘(𝑆‘𝐵))) | |
10 | 9 | breq2d 5160 | . . . . . . . . 9 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) ↔ (♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)))) |
11 | 10 | imbi1d 341 | . . . . . . . 8 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))) |
12 | 11 | 2ralbidv 3217 | . . . . . . 7 ⊢ ((𝑆‘𝐴) = (𝑆‘𝐵) → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))) |
13 | 8, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))) |
14 | 7, 13 | mpbid 231 | . . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐵)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) |
15 | efgredlem.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) | |
16 | efgredlem.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) | |
17 | 8 | eqcomd 2737 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑆‘𝐴)) |
18 | efgredlem.5 | . . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) | |
19 | eqcom 2738 | . . . . . 6 ⊢ ((𝐴‘0) = (𝐵‘0) ↔ (𝐵‘0) = (𝐴‘0)) | |
20 | 18, 19 | sylnib 328 | . . . . 5 ⊢ (𝜑 → ¬ (𝐵‘0) = (𝐴‘0)) |
21 | efgredlemb.l | . . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) | |
22 | efgredlemb.k | . . . . 5 ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) | |
23 | efgredlemb.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) | |
24 | efgredlemb.p | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) | |
25 | efgredlemb.v | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) | |
26 | efgredlemb.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) | |
27 | efgredlemb.7 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) | |
28 | efgredlemb.6 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) | |
29 | efgredlemb.8 | . . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) | |
30 | eqcom 2738 | . . . . . 6 ⊢ ((𝐴‘𝐾) = (𝐵‘𝐿) ↔ (𝐵‘𝐿) = (𝐴‘𝐾)) | |
31 | 29, 30 | sylnib 328 | . . . . 5 ⊢ (𝜑 → ¬ (𝐵‘𝐿) = (𝐴‘𝐾)) |
32 | 1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31 | efgredlemc 19661 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℤ≥‘𝑃) → (𝐵‘0) = (𝐴‘0))) |
33 | 32, 19 | imbitrrdi 251 | . . 3 ⊢ (𝜑 → (𝑄 ∈ (ℤ≥‘𝑃) → (𝐴‘0) = (𝐵‘0))) |
34 | 1, 2, 3, 4, 5, 6, 7, 16, 15, 8, 18, 22, 21, 24, 23, 26, 25, 28, 27, 29 | efgredlemc 19661 | . . 3 ⊢ (𝜑 → (𝑃 ∈ (ℤ≥‘𝑄) → (𝐴‘0) = (𝐵‘0))) |
35 | 24 | elfzelzd 13509 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℤ) |
36 | 23 | elfzelzd 13509 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℤ) |
37 | uztric 12853 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑄 ∈ (ℤ≥‘𝑃) ∨ 𝑃 ∈ (ℤ≥‘𝑄))) | |
38 | 35, 36, 37 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑄 ∈ (ℤ≥‘𝑃) ∨ 𝑃 ∈ (ℤ≥‘𝑄))) |
39 | 33, 34, 38 | mpjaod 857 | . 2 ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0)) |
40 | 39, 18 | pm2.65i 193 | 1 ⊢ ¬ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 ∖ cdif 3945 ∅c0 4322 {csn 4628 〈cop 4634 〈cotp 4636 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 × cxp 5674 dom cdm 5676 ran crn 5677 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 1oc1o 8465 2oc2o 8466 0cc0 11116 1c1 11117 < clt 11255 − cmin 11451 ℤcz 12565 ℤ≥cuz 12829 ...cfz 13491 ..^cfzo 13634 ♯chash 14297 Word cword 14471 splice csplice 14706 〈“cs2 14799 ~FG cefg 19622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 df-substr 14598 df-pfx 14628 df-splice 14707 df-s2 14806 |
This theorem is referenced by: efgredlem 19663 |
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