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Theorem krippen 29021

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 486	. . 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 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐴 ∈ 𝑃)
12		krippen.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐵 ∈ 𝑃)
14		krippen.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ 𝑃)
16		krippen.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
17	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐸 ∈ 𝑃)
18		krippen.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐹 ∈ 𝑃)
20		krippen.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
21	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝑋 ∈ 𝑃)
22		krippen.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
23	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝑌 ∈ 𝑃)
24		krippen.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))
25	24	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝐴𝐼𝐸))
26		krippen.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))
27	26	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝐵𝐼𝐹))
28		krippen.3	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))
29	28	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
30		krippen.4	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))
31	30	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
32		krippen.5	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))
33	32	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐵 = (𝑀‘𝐴))
34		krippen.6	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))
35	34	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐹 = (𝑁‘𝐸))
36		eqid 2765	. . 3 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
37		simpr 490	. . 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 29020	. 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸)) → 𝐶 ∈ (𝑋𝐼𝑌))
39	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐺 ∈ TarskiG)
40	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝑌 ∈ 𝑃)
41	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ 𝑃)
42	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝑋 ∈ 𝑃)
43	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐸 ∈ 𝑃)
44	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐹 ∈ 𝑃)
45	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐴 ∈ 𝑃)
46	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐵 ∈ 𝑃)
47	24	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐴𝐼𝐸))
48	1, 2, 3, 39, 45, 41, 43, 47	tgbtwncom 28810	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐸𝐼𝐴))
49	26	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐵𝐼𝐹))
50	1, 2, 3, 39, 46, 41, 44, 49	tgbtwncom 28810	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝐹𝐼𝐵))
51	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐸) = (𝐶 − 𝐹))
52	28	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → (𝐶 − 𝐴) = (𝐶 − 𝐵))
53	34	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐹 = (𝑁‘𝐸))
54	32	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐵 = (𝑀‘𝐴))
55		simpr 490	. . . 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 29020	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑌𝐼𝑋))
57	1, 2, 3, 39, 40, 41, 42, 56	tgbtwncom 28810	. 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑋𝐼𝑌))
58	1, 2, 3, 36, 6, 14, 10, 14, 16	legtrid 28913	. 2 ⊢ (𝜑 → ((𝐶 − 𝐴)(≤G‘𝐺)(𝐶 − 𝐸) ∨ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)))
59	38, 57, 58	mpjaodan 973	1 ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 Basecbs 17293 distcds 17343 TarskiGcstrkg 28749 Itvcitv 28755 LineGclng 28756 ≤Gcleg 28904 pInvGcmir 28982

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-xnn0 12595 df-z 12609 df-uz 12881 df-fz 13554 df-fzo 13702 df-hash 14387 df-word 14571 df-concat 14628 df-s1 14655 df-s2 14911 df-s3 14912 df-trkgc 28770 df-trkgb 28771 df-trkgcb 28772 df-trkg 28775 df-cgrg 28833 df-leg 28905 df-mir 28983

This theorem is used by: footexALT 29051 footexlem1 29052 mideulem2 29068

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