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Theorem krippen 27633

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.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
krippen	⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))

**Proof of Theorem krippen**
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 481	. . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐴 ∈ 𝑃)
12		krippen.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐵 ∈ 𝑃)
14		krippen.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ 𝑃)
16		krippen.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
17	16	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐸 ∈ 𝑃)
18		krippen.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
19	18	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐹 ∈ 𝑃)
20		krippen.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
21	20	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝑋 ∈ 𝑃)
22		krippen.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
23	22	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝑌 ∈ 𝑃)
24		krippen.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))
25	24	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝐴𝐼𝐸))
26		krippen.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))
27	26	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝐵𝐼𝐹))
28		krippen.3	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))
29	28	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
30		krippen.4	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))
31	30	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
32		krippen.5	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))
33	32	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐵 = (𝑀‘𝐴))
34		krippen.6	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))
35	34	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐹 = (𝑁‘𝐸))
36		eqid 2736	. . 3 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
37		simpr 485	. . 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 27632	. 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝑋𝐼𝑌))
39	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐺 ∈ TarskiG)
40	22	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝑌 ∈ 𝑃)
41	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ 𝑃)
42	20	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝑋 ∈ 𝑃)
43	16	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐸 ∈ 𝑃)
44	18	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐹 ∈ 𝑃)
45	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐴 ∈ 𝑃)
46	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐵 ∈ 𝑃)
47	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐴𝐼𝐸))
48	1, 2, 3, 39, 45, 41, 43, 47	tgbtwncom 27430	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐸𝐼𝐴))
49	26	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐵𝐼𝐹))
50	1, 2, 3, 39, 46, 41, 44, 49	tgbtwncom 27430	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐹𝐼𝐵))
51	30	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
52	28	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
53	34	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐹 = (𝑁‘𝐸))
54	32	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐵 = (𝑀‘𝐴))
55		simpr 485	. . . 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 27632	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑌𝐼𝑋))
57	1, 2, 3, 39, 40, 41, 42, 56	tgbtwncom 27430	. 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑋𝐼𝑌))
58	1, 2, 3, 36, 6, 14, 10, 14, 16	legtrid 27533	. 2 ⊢ (𝜑 → ((𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸) ∨ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)))
59	38, 57, 58	mpjaodan 957	1 ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 distcds 17142 TarskiGcstrkg 27369 Itvcitv 27375 LineGclng 27376 ≤Gcleg 27524 pInvGcmir 27594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-s2 14737 df-s3 14738 df-trkgc 27390 df-trkgb 27391 df-trkgcb 27392 df-trkg 27395 df-cgrg 27453 df-leg 27525 df-mir 27595

This theorem is referenced by: footexALT 27660 footexlem1 27661 mideulem2 27676

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