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Theorem mideu 29220

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 29219	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
11	5	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐺 ∈ TarskiG)
12		simplrl 789	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 ∈ 𝑃)
13		simplrr 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑦 ∈ 𝑃)
14	7	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐴 ∈ 𝑃)
15	8	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 ∈ 𝑃)
16		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
17	16	eqcomd 2766	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → ((𝑆‘𝑥)‘𝐴) = 𝐵)
18		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑦)‘𝐴))
19	18	eqcomd 2766	. . . . . 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 29163	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 = 𝑦)
21	20	ex 418	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
22	21	ralrimivva 3205	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
23		fveq2 6886	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑆‘𝑥) = (𝑆‘𝑦))
24	23	fveq1d 6888	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑆‘𝑥)‘𝐴) = ((𝑆‘𝑦)‘𝐴))
25	24	eqeq2d 2771	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑦)‘𝐴)))
26	25	rmo4 3687	. . 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 237	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
28		reu5 3367	. 2 ⊢ (∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ (∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)))
29	10, 27, 28	sylanbrc 595	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 ∃*wrmo 3364 class class class wbr 5102 ‘cfv 6539 2c2 12401 Basecbs 17393 distcds 17443 TarskiGcstrkg 28895 Dim_TarskiG≥cstrkgld 28899 Itvcitv 28901 LineGclng 28902 pInvGcmir 29130

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 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

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 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 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 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-oadd 8484 df-er 8721 df-map 8853 df-pm 8854 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-dju 9986 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-n0 12611 df-xnn0 12684 df-z 12698 df-uz 12970 df-fz 13644 df-fzo 13793 df-hash 14479 df-word 14663 df-concat 14720 df-s1 14747 df-s2 15004 df-s3 15005 df-trkgc 28916 df-trkgb 28917 df-trkgcb 28918 df-trkgld 28920 df-trkg 28921 df-cgrg 28980 df-leg 29052 df-mir 29131 df-rag 29175 df-perpg 29177

This theorem is used by: midf 29287 ismidb 29289

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