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Theorem mideu 28820

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 28819	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
11	5	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐺 ∈ TarskiG)
12		simplrl 777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 ∈ 𝑃)
13		simplrr 778	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑦 ∈ 𝑃)
14	7	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐴 ∈ 𝑃)
15	8	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 ∈ 𝑃)
16		simprl 771	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
17	16	eqcomd 2743	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → ((𝑆‘𝑥)‘𝐴) = 𝐵)
18		simprr 773	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑦)‘𝐴))
19	18	eqcomd 2743	. . . . . 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 28767	. . . . 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 3181	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
23		fveq2 6834	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑆‘𝑥) = (𝑆‘𝑦))
24	23	fveq1d 6836	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑆‘𝑥)‘𝐴) = ((𝑆‘𝑦)‘𝐴))
25	24	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑦)‘𝐴)))
26	25	rmo4 3677	. . 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 234	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
28		reu5 3345	. 2 ⊢ (∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ (∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)))
29	10, 27, 28	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∃!wreu 3341 ∃*wrmo 3342 class class class wbr 5086 ‘cfv 6492 2c2 12227 Basecbs 17170 distcds 17220 TarskiGcstrkg 28509 Dim_TarskiG≥cstrkgld 28513 Itvcitv 28515 LineGclng 28516 pInvGcmir 28734

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-oadd 8402 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-concat 14524 df-s1 14550 df-s2 14801 df-s3 14802 df-trkgc 28530 df-trkgb 28531 df-trkgcb 28532 df-trkgld 28534 df-trkg 28535 df-cgrg 28593 df-leg 28665 df-mir 28735 df-rag 28776 df-perpg 28778

This theorem is referenced by: midf 28858 ismidb 28860

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