![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem1 | Structured version Visualization version GIF version |
Description: Lemma for cycpmco2 33136. (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 3975 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
5 | ssrab2 4090 | . . . . . . . 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 2735 | . . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 1, 7, 8 | tocycf 33120 | . . . . . . . . . . 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 6747 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
12 | 6, 11 | eleqtrd 2841 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
13 | 5, 12 | sselid 3993 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
14 | id 22 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
15 | dmeq 5917 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊) | |
16 | eqidd 2736 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷) | |
17 | 14, 15, 16 | f1eq123d 6841 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
18 | 17 | elrab3 3696 | . . . . . . . 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 6805 | . . . . . 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 6745 | . . . 4 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷) |
24 | cycpmco2.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) | |
25 | 23, 24 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
26 | 3 | eldifbd 3976 | . . . . 5 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊) |
27 | nelne2 3038 | . . . . 5 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼) | |
28 | 24, 26, 27 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐼) |
29 | 28 | necomd 2994 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 33124 | . 2 ⊢ (𝜑 → ((𝑀‘〈“𝐼𝐽”〉)‘𝐼) = 𝐽) |
31 | 30 | fveq2d 6911 | 1 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 ∖ cdif 3960 〈cotp 4639 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ⟶wf 6559 –1-1→wf1 6560 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 Word cword 14549 〈“cs1 14630 splice csplice 14784 〈“cs2 14877 Basecbs 17245 SymGrpcsymg 19401 toCycctocyc 33109 |
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 ax-pre-sup 11231 |
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-rmo 3378 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-tp 4636 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-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 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-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-csh 14824 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-tset 17317 df-efmnd 18895 df-symg 19402 df-tocyc 33110 |
This theorem is referenced by: cycpmco2lem4 33132 |
Copyright terms: Public domain | W3C validator |