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Theorem clwlkclwwlkflem 29257
Description: Lemma for clwlkclwwlkf 29261. (Contributed by AV, 24-May-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
clwlkclwwlkf.a 𝐴 = (1st β€˜π‘ˆ)
clwlkclwwlkf.b 𝐡 = (2nd β€˜π‘ˆ)
Assertion
Ref Expression
clwlkclwwlkflem (π‘ˆ ∈ 𝐢 β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))
Distinct variable groups:   𝑀,𝐺   𝑀,𝐴   𝑀,π‘ˆ
Allowed substitution hints:   𝐡(𝑀)   𝐢(𝑀)

Proof of Theorem clwlkclwwlkflem
StepHypRef Expression
1 fveq2 6892 . . . . . 6 (𝑀 = π‘ˆ β†’ (1st β€˜π‘€) = (1st β€˜π‘ˆ))
2 clwlkclwwlkf.a . . . . . 6 𝐴 = (1st β€˜π‘ˆ)
31, 2eqtr4di 2791 . . . . 5 (𝑀 = π‘ˆ β†’ (1st β€˜π‘€) = 𝐴)
43fveq2d 6896 . . . 4 (𝑀 = π‘ˆ β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜π΄))
54breq2d 5161 . . 3 (𝑀 = π‘ˆ β†’ (1 ≀ (β™―β€˜(1st β€˜π‘€)) ↔ 1 ≀ (β™―β€˜π΄)))
6 clwlkclwwlkf.c . . 3 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
75, 6elrab2 3687 . 2 (π‘ˆ ∈ 𝐢 ↔ (π‘ˆ ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π΄)))
8 clwlkwlk 29032 . . . 4 (π‘ˆ ∈ (ClWalksβ€˜πΊ) β†’ π‘ˆ ∈ (Walksβ€˜πΊ))
9 wlkop 28885 . . . . 5 (π‘ˆ ∈ (Walksβ€˜πΊ) β†’ π‘ˆ = ⟨(1st β€˜π‘ˆ), (2nd β€˜π‘ˆ)⟩)
10 clwlkclwwlkf.b . . . . . . . 8 𝐡 = (2nd β€˜π‘ˆ)
112, 10opeq12i 4879 . . . . . . 7 ⟨𝐴, 𝐡⟩ = ⟨(1st β€˜π‘ˆ), (2nd β€˜π‘ˆ)⟩
1211eqeq2i 2746 . . . . . 6 (π‘ˆ = ⟨𝐴, 𝐡⟩ ↔ π‘ˆ = ⟨(1st β€˜π‘ˆ), (2nd β€˜π‘ˆ)⟩)
13 eleq1 2822 . . . . . . 7 (π‘ˆ = ⟨𝐴, 𝐡⟩ β†’ (π‘ˆ ∈ (ClWalksβ€˜πΊ) ↔ ⟨𝐴, 𝐡⟩ ∈ (ClWalksβ€˜πΊ)))
14 df-br 5150 . . . . . . . 8 (𝐴(ClWalksβ€˜πΊ)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ (ClWalksβ€˜πΊ))
15 isclwlk 29030 . . . . . . . . 9 (𝐴(ClWalksβ€˜πΊ)𝐡 ↔ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))))
16 wlkcl 28872 . . . . . . . . . . . . . . 15 (𝐴(Walksβ€˜πΊ)𝐡 β†’ (β™―β€˜π΄) ∈ β„•0)
17 elnnnn0c 12517 . . . . . . . . . . . . . . . 16 ((β™―β€˜π΄) ∈ β„• ↔ ((β™―β€˜π΄) ∈ β„•0 ∧ 1 ≀ (β™―β€˜π΄)))
1817a1i 11 . . . . . . . . . . . . . . 15 (𝐴(Walksβ€˜πΊ)𝐡 β†’ ((β™―β€˜π΄) ∈ β„• ↔ ((β™―β€˜π΄) ∈ β„•0 ∧ 1 ≀ (β™―β€˜π΄))))
1916, 18mpbirand 706 . . . . . . . . . . . . . 14 (𝐴(Walksβ€˜πΊ)𝐡 β†’ ((β™―β€˜π΄) ∈ β„• ↔ 1 ≀ (β™―β€˜π΄)))
2019bicomd 222 . . . . . . . . . . . . 13 (𝐴(Walksβ€˜πΊ)𝐡 β†’ (1 ≀ (β™―β€˜π΄) ↔ (β™―β€˜π΄) ∈ β„•))
2120adantr 482 . . . . . . . . . . . 12 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))) β†’ (1 ≀ (β™―β€˜π΄) ↔ (β™―β€˜π΄) ∈ β„•))
2221pm5.32i 576 . . . . . . . . . . 11 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))) ∧ 1 ≀ (β™―β€˜π΄)) ↔ ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))) ∧ (β™―β€˜π΄) ∈ β„•))
23 df-3an 1090 . . . . . . . . . . 11 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ↔ ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))) ∧ (β™―β€˜π΄) ∈ β„•))
2422, 23sylbb2 237 . . . . . . . . . 10 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))) ∧ 1 ≀ (β™―β€˜π΄)) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))
2524ex 414 . . . . . . . . 9 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))) β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•)))
2615, 25sylbi 216 . . . . . . . 8 (𝐴(ClWalksβ€˜πΊ)𝐡 β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•)))
2714, 26sylbir 234 . . . . . . 7 (⟨𝐴, 𝐡⟩ ∈ (ClWalksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•)))
2813, 27syl6bi 253 . . . . . 6 (π‘ˆ = ⟨𝐴, 𝐡⟩ β†’ (π‘ˆ ∈ (ClWalksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))))
2912, 28sylbir 234 . . . . 5 (π‘ˆ = ⟨(1st β€˜π‘ˆ), (2nd β€˜π‘ˆ)⟩ β†’ (π‘ˆ ∈ (ClWalksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))))
309, 29syl 17 . . . 4 (π‘ˆ ∈ (Walksβ€˜πΊ) β†’ (π‘ˆ ∈ (ClWalksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))))
318, 30mpcom 38 . . 3 (π‘ˆ ∈ (ClWalksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜π΄) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•)))
3231imp 408 . 2 ((π‘ˆ ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π΄)) β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))
337, 32sylbi 216 1 (π‘ˆ ∈ 𝐢 β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  βŸ¨cop 4635   class class class wbr 5149  β€˜cfv 6544  1st c1st 7973  2nd c2nd 7974  0cc0 11110  1c1 11111   ≀ cle 11249  β„•cn 12212  β„•0cn0 12472  β™―chash 14290  Walkscwlks 28853  ClWalkscclwlks 29027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-wlks 28856  df-clwlks 29028
This theorem is referenced by:  clwlkclwwlkf1lem2  29258  clwlkclwwlkf1lem3  29259  clwlkclwwlkf  29261  clwlkclwwlkf1  29263
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