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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for cycpmco2 33154. (Contributed by Thierry Arnoux, 4-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| cycpmco2.c | ⊢ 𝑀 = (toCyc‘𝐷) | 
| cycpmco2.s | ⊢ 𝑆 = (SymGrp‘𝐷) | 
| cycpmco2.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) | 
| cycpmco2.w | ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) | 
| cycpmco2.i | ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) | 
| cycpmco2.j | ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) | 
| cycpmco2.e | ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) | 
| cycpmco2.1 | ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) | 
| Ref | Expression | 
|---|---|
| cycpmco2lem1 | ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cycpmco2.c | . . 3 ⊢ 𝑀 = (toCyc‘𝐷) | |
| 2 | cycpmco2.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
| 3 | cycpmco2.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) | |
| 4 | 3 | eldifad 3962 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | 
| 5 | ssrab2 4079 | . . . . . . . 8 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷 | |
| 6 | cycpmco2.w | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) | |
| 7 | cycpmco2.s | . . . . . . . . . . . 12 ⊢ 𝑆 = (SymGrp‘𝐷) | |
| 8 | eqid 2736 | . . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 9 | 1, 7, 8 | tocycf 33138 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) | 
| 10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) | 
| 11 | 10 | fdmd 6745 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) | 
| 12 | 6, 11 | eleqtrd 2842 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) | 
| 13 | 5, 12 | sselid 3980 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) | 
| 14 | id 22 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
| 15 | dmeq 5913 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊) | |
| 16 | eqidd 2737 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷) | |
| 17 | 14, 15, 16 | f1eq123d 6839 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) | 
| 18 | 17 | elrab3 3692 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ 𝑊:dom 𝑊–1-1→𝐷)) | 
| 19 | 18 | biimpa 476 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) → 𝑊:dom 𝑊–1-1→𝐷) | 
| 20 | 13, 12, 19 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) | 
| 21 | f1f 6803 | . . . . . 6 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊⟶𝐷) | |
| 22 | 20, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝐷) | 
| 23 | 22 | frnd 6743 | . . . 4 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷) | 
| 24 | cycpmco2.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) | |
| 25 | 23, 24 | sseldd 3983 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | 
| 26 | 3 | eldifbd 3963 | . . . . 5 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊) | 
| 27 | nelne2 3039 | . . . . 5 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼) | |
| 28 | 24, 26, 27 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐼) | 
| 29 | 28 | necomd 2995 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | 
| 30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 33142 | . 2 ⊢ (𝜑 → ((𝑀‘〈“𝐼𝐽”〉)‘𝐼) = 𝐽) | 
| 31 | 30 | fveq2d 6909 | 1 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {crab 3435 ∖ cdif 3947 〈cotp 4633 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ⟶wf 6556 –1-1→wf1 6557 ‘cfv 6560 (class class class)co 7432 1c1 11157 + caddc 11159 Word cword 14553 〈“cs1 14634 splice csplice 14788 〈“cs2 14881 Basecbs 17248 SymGrpcsymg 19387 toCycctocyc 33127 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-substr 14680 df-pfx 14710 df-csh 14828 df-s2 14888 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-tset 17317 df-efmnd 18883 df-symg 19388 df-tocyc 33128 | 
| This theorem is referenced by: cycpmco2lem4 33150 | 
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