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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 2737 | . . 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 28698 | . 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 28496 | . . . 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 28496 | . . . 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 28698 | . . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑌𝐼𝑋)) |
| 57 | 1, 2, 3, 39, 40, 41, 42, 56 | tgbtwncom 28496 | . 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑋𝐼𝑌)) |
| 58 | 1, 2, 3, 36, 6, 14, 10, 14, 16 | legtrid 28599 | . 2 ⊢ (𝜑 → ((𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸) ∨ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴))) |
| 59 | 38, 57, 58 | mpjaodan 961 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 ≤Gcleg 28590 pInvGcmir 28660 |
| 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-rmo 3380 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-tp 4631 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-oadd 8510 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 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-3 12330 df-n0 12527 df-xnn0 12600 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 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-cgrg 28519 df-leg 28591 df-mir 28661 |
| This theorem is referenced by: footexALT 28726 footexlem1 28727 mideulem2 28742 |
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