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Mirrors > Home > MPE Home > Th. List > krippen | Structured version Visualization version GIF version |
Description: Krippenlemma (German for crib's lemma) Lemma 7.22 of [Schwabhauser] p. 53. proven by Gupta 1965 as Theorem 3.45. (Contributed by Thierry Arnoux, 12-Aug-2019.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
krippen.m | ⊢ 𝑀 = (𝑆‘𝑋) |
krippen.n | ⊢ 𝑁 = (𝑆‘𝑌) |
krippen.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
krippen.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
krippen.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
krippen.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
krippen.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
krippen.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
krippen.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
krippen.1 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸)) |
krippen.2 | ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹)) |
krippen.3 | ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
krippen.4 | ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹)) |
krippen.5 | ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴)) |
krippen.6 | ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸)) |
Ref | Expression |
---|---|
krippen | ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐺 ∈ TarskiG) |
8 | krippen.m | . . 3 ⊢ 𝑀 = (𝑆‘𝑋) | |
9 | krippen.n | . . 3 ⊢ 𝑁 = (𝑆‘𝑌) | |
10 | krippen.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐴 ∈ 𝑃) |
12 | krippen.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐵 ∈ 𝑃) |
14 | krippen.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
15 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ 𝑃) |
16 | krippen.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
17 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐸 ∈ 𝑃) |
18 | krippen.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
19 | 18 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐹 ∈ 𝑃) |
20 | krippen.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
21 | 20 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝑋 ∈ 𝑃) |
22 | krippen.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
23 | 22 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝑌 ∈ 𝑃) |
24 | krippen.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸)) | |
25 | 24 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝐴𝐼𝐸)) |
26 | krippen.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹)) | |
27 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝐵𝐼𝐹)) |
28 | krippen.3 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) | |
29 | 28 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
30 | krippen.4 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹)) | |
31 | 30 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐸) = (𝐶 − 𝐹)) |
32 | krippen.5 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴)) | |
33 | 32 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐵 = (𝑀‘𝐴)) |
34 | krippen.6 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸)) | |
35 | 34 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐹 = (𝑁‘𝐸)) |
36 | eqid 2734 | . . 3 ⊢ (≤G‘𝐺) = (≤G‘𝐺) | |
37 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) | |
38 | 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 36, 37 | krippenlem 28712 | . 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝑋𝐼𝑌)) |
39 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐺 ∈ TarskiG) |
40 | 22 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝑌 ∈ 𝑃) |
41 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ 𝑃) |
42 | 20 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝑋 ∈ 𝑃) |
43 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐸 ∈ 𝑃) |
44 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐹 ∈ 𝑃) |
45 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐴 ∈ 𝑃) |
46 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐵 ∈ 𝑃) |
47 | 24 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐴𝐼𝐸)) |
48 | 1, 2, 3, 39, 45, 41, 43, 47 | tgbtwncom 28510 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐸𝐼𝐴)) |
49 | 26 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐵𝐼𝐹)) |
50 | 1, 2, 3, 39, 46, 41, 44, 49 | tgbtwncom 28510 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐹𝐼𝐵)) |
51 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐸) = (𝐶 − 𝐹)) |
52 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
53 | 34 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐹 = (𝑁‘𝐸)) |
54 | 32 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐵 = (𝑀‘𝐴)) |
55 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) | |
56 | 1, 2, 3, 4, 5, 39, 9, 8, 43, 44, 41, 45, 46, 40, 42, 48, 50, 51, 52, 53, 54, 36, 55 | krippenlem 28712 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑌𝐼𝑋)) |
57 | 1, 2, 3, 39, 40, 41, 42, 56 | tgbtwncom 28510 | . 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑋𝐼𝑌)) |
58 | 1, 2, 3, 36, 6, 14, 10, 14, 16 | legtrid 28613 | . 2 ⊢ (𝜑 → ((𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸) ∨ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴))) |
59 | 38, 57, 58 | mpjaodan 960 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 distcds 17306 TarskiGcstrkg 28449 Itvcitv 28455 LineGclng 28456 ≤Gcleg 28604 pInvGcmir 28674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-trkgc 28470 df-trkgb 28471 df-trkgcb 28472 df-trkg 28475 df-cgrg 28533 df-leg 28605 df-mir 28675 |
This theorem is referenced by: footexALT 28740 footexlem1 28741 mideulem2 28756 |
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