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Theorem clwlkclwwlkflem 28997
Description: Lemma for clwlkclwwlkf 29001. (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 6846 . . . . . 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 6850 . . . 4 (𝑀 = π‘ˆ β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜π΄))
54breq2d 5121 . . 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 3652 . 2 (π‘ˆ ∈ 𝐢 ↔ (π‘ˆ ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π΄)))
8 clwlkwlk 28772 . . . 4 (π‘ˆ ∈ (ClWalksβ€˜πΊ) β†’ π‘ˆ ∈ (Walksβ€˜πΊ))
9 wlkop 28625 . . . . 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 4839 . . . . . . 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 5110 . . . . . . . 8 (𝐴(ClWalksβ€˜πΊ)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ (ClWalksβ€˜πΊ))
15 isclwlk 28770 . . . . . . . . 9 (𝐴(ClWalksβ€˜πΊ)𝐡 ↔ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄))))
16 wlkcl 28612 . . . . . . . . . . . . . . 15 (𝐴(Walksβ€˜πΊ)𝐡 β†’ (β™―β€˜π΄) ∈ β„•0)
17 elnnnn0c 12466 . . . . . . . . . . . . . . . 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 3406  βŸ¨cop 4596   class class class wbr 5109  β€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  0cc0 11059  1c1 11060   ≀ cle 11198  β„•cn 12161  β„•0cn0 12421  β™―chash 14239  Walkscwlks 28593  ClWalkscclwlks 28767
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-wlks 28596  df-clwlks 28768
This theorem is referenced by:  clwlkclwwlkf1lem2  28998  clwlkclwwlkf1lem3  28999  clwlkclwwlkf  29001  clwlkclwwlkf1  29003
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