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Theorem mideu 28894

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 28893	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
11	5	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐺 ∈ TarskiG)
12		simplrl 786	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 ∈ 𝑃)
13		simplrr 787	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑦 ∈ 𝑃)
14	7	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐴 ∈ 𝑃)
15	8	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 ∈ 𝑃)
16		simprl 780	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
17	16	eqcomd 2767	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → ((𝑆‘𝑥)‘𝐴) = 𝐵)
18		simprr 782	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝐵 = ((𝑆‘𝑦)‘𝐴))
19	18	eqcomd 2767	. . . . . 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 28841	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴))) → 𝑥 = 𝑦)
21	20	ex 416	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
22	21	ralrimivva 3204	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ((𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝐵 = ((𝑆‘𝑦)‘𝐴)) → 𝑥 = 𝑦))
23		fveq2 6861	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑆‘𝑥) = (𝑆‘𝑦))
24	23	fveq1d 6863	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑆‘𝑥)‘𝐴) = ((𝑆‘𝑦)‘𝐴))
25	24	eqeq2d 2772	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑦)‘𝐴)))
26	25	rmo4 3691	. . 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 236	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
28		reu5 3368	. 2 ⊢ (∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ↔ (∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ ∃*𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)))
29	10, 27, 28	sylanbrc 592	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ∃!wreu 3364 ∃*wrmo 3365 class class class wbr 5097 ‘cfv 6515 2c2 12265 Basecbs 17235 distcds 17285 TarskiGcstrkg 28583 Dim_TarskiG≥cstrkgld 28587 Itvcitv 28589 LineGclng 28590 pInvGcmir 28808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-oadd 8434 df-er 8671 df-map 8803 df-pm 8804 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-dju 9852 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-n0 12475 df-xnn0 12548 df-z 12562 df-uz 12833 df-fz 13506 df-fzo 13653 df-hash 14337 df-word 14520 df-concat 14577 df-s1 14603 df-s2 14854 df-s3 14855 df-trkgc 28604 df-trkgb 28605 df-trkgcb 28606 df-trkgld 28608 df-trkg 28609 df-cgrg 28667 df-leg 28739 df-mir 28809 df-rag 28850 df-perpg 28852

This theorem is referenced by: midf 28932 ismidb 28934

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