| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > wlk2v2elem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for wlk2v2e 30176: The values of 𝐼 after 𝐹 are edges between two vertices enumerated by 𝑃. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
| Ref | Expression |
|---|---|
| wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
| wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
| wlk2v2e.x | ⊢ 𝑋 ∈ V |
| wlk2v2e.y | ⊢ 𝑌 ∈ V |
| wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
| Ref | Expression |
|---|---|
| wlk2v2elem2 | ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
| 2 | 1 | fveq1i 6907 | . . . . . 6 ⊢ (𝐹‘0) = (〈“00”〉‘0) |
| 3 | 0z 12624 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
| 4 | s2fv0 14926 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘0) = 0) | |
| 5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘0) = 0 |
| 6 | 2, 5 | eqtri 2765 | . . . . 5 ⊢ (𝐹‘0) = 0 |
| 7 | 6 | fveq2i 6909 | . . . 4 ⊢ (𝐼‘(𝐹‘0)) = (𝐼‘0) |
| 8 | wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
| 9 | 8 | fveq1i 6907 | . . . . 5 ⊢ (𝐼‘0) = (〈“{𝑋, 𝑌}”〉‘0) |
| 10 | prex 5437 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
| 11 | s1fv 14648 | . . . . . 6 ⊢ ({𝑋, 𝑌} ∈ V → (〈“{𝑋, 𝑌}”〉‘0) = {𝑋, 𝑌}) | |
| 12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ (〈“{𝑋, 𝑌}”〉‘0) = {𝑋, 𝑌} |
| 13 | 9, 12 | eqtri 2765 | . . . 4 ⊢ (𝐼‘0) = {𝑋, 𝑌} |
| 14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
| 15 | 14 | fveq1i 6907 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝑋𝑌𝑋”〉‘0) |
| 16 | wlk2v2e.x | . . . . . . . 8 ⊢ 𝑋 ∈ V | |
| 17 | s3fv0 14930 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘0) = 𝑋) | |
| 18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘0) = 𝑋 |
| 19 | 15, 18 | eqtri 2765 | . . . . . 6 ⊢ (𝑃‘0) = 𝑋 |
| 20 | 14 | fveq1i 6907 | . . . . . . 7 ⊢ (𝑃‘1) = (〈“𝑋𝑌𝑋”〉‘1) |
| 21 | wlk2v2e.y | . . . . . . . 8 ⊢ 𝑌 ∈ V | |
| 22 | s3fv1 14931 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (〈“𝑋𝑌𝑋”〉‘1) = 𝑌) | |
| 23 | 21, 22 | ax-mp 5 | . . . . . . 7 ⊢ (〈“𝑋𝑌𝑋”〉‘1) = 𝑌 |
| 24 | 20, 23 | eqtri 2765 | . . . . . 6 ⊢ (𝑃‘1) = 𝑌 |
| 25 | 19, 24 | preq12i 4738 | . . . . 5 ⊢ {(𝑃‘0), (𝑃‘1)} = {𝑋, 𝑌} |
| 26 | 25 | eqcomi 2746 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘0), (𝑃‘1)} |
| 27 | 7, 13, 26 | 3eqtri 2769 | . . 3 ⊢ (𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} |
| 28 | 1 | fveq1i 6907 | . . . . . 6 ⊢ (𝐹‘1) = (〈“00”〉‘1) |
| 29 | s2fv1 14927 | . . . . . . 7 ⊢ (0 ∈ ℤ → (〈“00”〉‘1) = 0) | |
| 30 | 3, 29 | ax-mp 5 | . . . . . 6 ⊢ (〈“00”〉‘1) = 0 |
| 31 | 28, 30 | eqtri 2765 | . . . . 5 ⊢ (𝐹‘1) = 0 |
| 32 | 31 | fveq2i 6909 | . . . 4 ⊢ (𝐼‘(𝐹‘1)) = (𝐼‘0) |
| 33 | prcom 4732 | . . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
| 34 | 14 | fveq1i 6907 | . . . . . . . 8 ⊢ (𝑃‘2) = (〈“𝑋𝑌𝑋”〉‘2) |
| 35 | s3fv2 14932 | . . . . . . . . 9 ⊢ (𝑋 ∈ V → (〈“𝑋𝑌𝑋”〉‘2) = 𝑋) | |
| 36 | 16, 35 | ax-mp 5 | . . . . . . . 8 ⊢ (〈“𝑋𝑌𝑋”〉‘2) = 𝑋 |
| 37 | 34, 36 | eqtri 2765 | . . . . . . 7 ⊢ (𝑃‘2) = 𝑋 |
| 38 | 24, 37 | preq12i 4738 | . . . . . 6 ⊢ {(𝑃‘1), (𝑃‘2)} = {𝑌, 𝑋} |
| 39 | 38 | eqcomi 2746 | . . . . 5 ⊢ {𝑌, 𝑋} = {(𝑃‘1), (𝑃‘2)} |
| 40 | 33, 39 | eqtri 2765 | . . . 4 ⊢ {𝑋, 𝑌} = {(𝑃‘1), (𝑃‘2)} |
| 41 | 32, 13, 40 | 3eqtri 2769 | . . 3 ⊢ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} |
| 42 | 2wlklem 29685 | . . 3 ⊢ (∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) | |
| 43 | 27, 41, 42 | mpbir2an 711 | . 2 ⊢ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
| 44 | 14, 1 | 2wlkdlem2 29946 | . . 3 ⊢ (0..^(♯‘𝐹)) = {0, 1} |
| 45 | 44 | raleqi 3324 | . 2 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
| 46 | 43, 45 | mpbir 231 | 1 ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {cpr 4628 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 2c2 12321 ℤcz 12613 ..^cfzo 13694 ♯chash 14369 〈“cs1 14633 〈“cs2 14880 〈“cs3 14881 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 |
| This theorem is referenced by: wlk2v2e 30176 |
| Copyright terms: Public domain | W3C validator |