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Theorem krippen 29172

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 2760	. . 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 29171	. 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 28960	. . . 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 28960	. . . 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 29171	. . 3 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑌𝐼𝑋))
57	1, 2, 3, 39, 40, 41, 42, 56	tgbtwncom 28960	. 2 ⊢ ((𝜑 ∧ (𝐶 − 𝐸)(≤G‘𝐺)(𝐶 − 𝐴)) → 𝐶 ∈ (𝑋𝐼𝑌))
58	1, 2, 3, 36, 6, 14, 10, 14, 16	legtrid 29063	. 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 1569 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6543 (class class class)co 7424 Basecbs 17396 distcds 17446 TarskiGcstrkg 28898 Itvcitv 28904 LineGclng 28905 ≤Gcleg 29054 pInvGcmir 29133

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-oadd 8488 df-er 8725 df-pm 8858 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-dju 9989 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-n0 12614 df-xnn0 12687 df-z 12701 df-uz 12973 df-fz 13647 df-fzo 13796 df-hash 14482 df-word 14666 df-concat 14723 df-s1 14750 df-s2 15007 df-s3 15008 df-trkgc 28919 df-trkgb 28920 df-trkgcb 28921 df-trkg 28924 df-cgrg 28983 df-leg 29055 df-mir 29134

This theorem is used by: footexALT 29202 footexlem1 29203 mideulem2 29219

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