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Mirrors > Home > MPE Home > Th. List > efgredlemf | Structured version Visualization version GIF version |
Description: Lemma for efgredleme 19776. (Contributed by Mario Carneiro, 4-Jun-2016.) |
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) |
Ref | Expression |
---|---|
efgredlemf | ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgredlem.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) | |
2 | efgval.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
3 | efgval.r | . . . . . . . 8 ⊢ ∼ = ( ~FG ‘𝐼) | |
4 | efgval2.m | . . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
5 | efgval2.t | . . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
6 | efgred.d | . . . . . . . 8 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
7 | efgred.s | . . . . . . . 8 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | |
8 | 2, 3, 4, 5, 6, 7 | efgsdm 19763 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1))))) |
9 | 8 | simp1bi 1144 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
11 | 10 | eldifad 3975 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊) |
12 | wrdf 14554 | . . . 4 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) |
14 | fzossfz 13715 | . . . . 5 ⊢ (0..^((♯‘𝐴) − 1)) ⊆ (0...((♯‘𝐴) − 1)) | |
15 | lencl 14568 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ0) | |
16 | 11, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
17 | 16 | nn0zd 12637 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ) |
18 | fzoval 13697 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) |
20 | 14, 19 | sseqtrrid 4049 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝐴) − 1)) ⊆ (0..^(♯‘𝐴))) |
21 | efgredlemb.k | . . . . 5 ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) | |
22 | efgredlem.1 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) | |
23 | efgredlem.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) | |
24 | efgredlem.4 | . . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) | |
25 | efgredlem.5 | . . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) | |
26 | 2, 3, 4, 5, 6, 7, 22, 1, 23, 24, 25 | efgredlema 19773 | . . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ)) |
27 | 26 | simpld 494 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ) |
28 | fzo0end 13794 | . . . . . 6 ⊢ (((♯‘𝐴) − 1) ∈ ℕ → (((♯‘𝐴) − 1) − 1) ∈ (0..^((♯‘𝐴) − 1))) | |
29 | 27, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐴) − 1) − 1) ∈ (0..^((♯‘𝐴) − 1))) |
30 | 21, 29 | eqeltrid 2843 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1))) |
31 | 20, 30 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴))) |
32 | 13, 31 | ffvelcdmd 7105 | . 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
33 | 2, 3, 4, 5, 6, 7 | efgsdm 19763 | . . . . . . 7 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1))))) |
34 | 33 | simp1bi 1144 | . . . . . 6 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
35 | 23, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
36 | 35 | eldifad 3975 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊) |
37 | wrdf 14554 | . . . 4 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) |
39 | fzossfz 13715 | . . . . 5 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1)) | |
40 | lencl 14568 | . . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ0) | |
41 | 36, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
42 | 41 | nn0zd 12637 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ) |
43 | fzoval 13697 | . . . . . 6 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) | |
44 | 42, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) |
45 | 39, 44 | sseqtrrid 4049 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝐵) − 1)) ⊆ (0..^(♯‘𝐵))) |
46 | efgredlemb.l | . . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) | |
47 | fzo0end 13794 | . . . . . 6 ⊢ (((♯‘𝐵) − 1) ∈ ℕ → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) | |
48 | 26, 47 | simpl2im 503 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) |
49 | 46, 48 | eqeltrid 2843 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1))) |
50 | 45, 49 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵))) |
51 | 38, 50 | ffvelcdmd 7105 | . 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
52 | 32, 51 | jca 511 | 1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∖ cdif 3960 ∅c0 4339 {csn 4631 〈cop 4637 〈cotp 4639 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 I cid 5582 × cxp 5687 dom cdm 5689 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1oc1o 8498 2oc2o 8499 0cc0 11153 1c1 11154 < clt 11293 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 splice csplice 14784 〈“cs2 14877 ~FG cefg 19739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 |
This theorem is referenced by: efgredlemg 19775 efgredleme 19776 |
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