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Theorem mideu 27003

Description: Existence and uniqueness of the midpoint, Theorem 8.22 of [Schwabhauser] p. 64. (Contributed by Thierry Arnoux, 25-Nov-2019.)

**Hypotheses**
Ref	Expression
colperpex.p	⊢ 𝑃 = (Base‘𝐺)
colperpex.d	⊢ − = (dist‘𝐺)
colperpex.i	⊢ 𝐼 = (Itv‘𝐺)
colperpex.l	⊢ 𝐿 = (LineG‘𝐺)
colperpex.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mideu.s	⊢ 𝑆 = (pInvG‘𝐺)
mideu.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mideu.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mideu.3	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)

**Assertion**
Ref	Expression
mideu	⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Distinct variable groups: 𝑥, − 𝑥,𝐴 𝑥,𝐵 𝑥,𝐺 𝑥,𝐼 𝑥,𝐿 𝑥,𝑃 𝑥,𝑆 𝜑,𝑥

Proof of Theorem mideu Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		colperpex.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		colperpex.d	. . 3 ⊢ − = (dist‘𝐺)
3		colperpex.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		colperpex.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		colperpex.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		mideu.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
7		mideu.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mideu.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9		mideu.3	. . 3 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	midex 27002	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
11	5	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐺 ∈ TarskiG)
12		simplrl 773	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 ∈ 𝑃)
13		simplrr 774	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑦 ∈ 𝑃)
14	7	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐴 ∈ 𝑃)
15	8	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 ∈ 𝑃)
16		simprl 767	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
17	16	eqcomd 2744	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → ((𝑆‘𝑥)‘𝐴) = 𝐵)
18		simprr 769	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑦)‘𝐴))
19	18	eqcomd 2744	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → ((𝑆‘𝑦)‘𝐴) = 𝐵)
20	1, 2, 3, 4, 6, 11, 12, 13, 14, 15, 17, 19	miduniq 26950	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 = 𝑦)
21	20	ex 412	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
22	21	ralrimivva 3114	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
23		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑆‘𝑥) = (𝑆‘𝑦))
24	23	fveq1d 6758	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑆‘𝑥)‘𝐴) = ((𝑆‘𝑦)‘𝐴))
25	24	eqeq2d 2749	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑦)‘𝐴)))
26	25	rmo4 3660	. . 3 ⊢ (∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
27	22, 26	sylibr 233	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
28		reu5 3351	. 2 ⊢ (∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ (∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)))
29	10, 27, 28	sylanbrc 582	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ∃!wreu 3065 ∃*wrmo 3066 class class class wbr 5070 ‘cfv 6418 2c2 11958 Basecbs 16840 distcds 16897 TarskiGcstrkg 26693 Dim_TarskiG≥cstrkgld 26697 Itvcitv 26699 LineGclng 26700 pInvGcmir 26917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-s2 14489 df-s3 14490 df-trkgc 26713 df-trkgb 26714 df-trkgcb 26715 df-trkgld 26717 df-trkg 26718 df-cgrg 26776 df-leg 26848 df-mir 26918 df-rag 26959 df-perpg 26961

This theorem is referenced by: midf 27041 ismidb 27043

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