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Theorem mideulem 27093

Description: Lemma for mideu 27095. We can assume mideulem.9 "without loss of generality". (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)
mideulem.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
mideulem.2	⊢ (𝜑 → 𝑄 ∈ 𝑃)
mideulem.3	⊢ (𝜑 → 𝑂 ∈ 𝑃)
mideulem.4	⊢ (𝜑 → 𝑇 ∈ 𝑃)
mideulem.5	⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
mideulem.6	⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
mideulem.7	⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
mideulem.8	⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
mideulem.9	⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄))

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

Distinct variable groups: 𝑥, − 𝑥,𝐴 𝑥,𝐵 𝑥,𝐼 𝑥,𝑂 𝑥,𝑃 𝑥,𝑄 𝑥,𝑇 𝜑,𝑥 Allowed substitution hints: 𝑆(𝑥) 𝐺(𝑥) 𝐿(𝑥)

Proof of Theorem mideulem Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprrl 778	. . 3 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) ∧ (𝑥 ∈ 𝑃 ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
2		colperpex.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		colperpex.d	. . . 4 ⊢ − = (dist‘𝐺)
4		colperpex.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		colperpex.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
6		colperpex.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐺 ∈ TarskiG)
8		mideu.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
9		mideu.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10	9	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ∈ 𝑃)
11		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐵 ∈ 𝑃)
13		mideulem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
14	13	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ≠ 𝐵)
15		mideulem.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
16	15	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑄 ∈ 𝑃)
17		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
18	17	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑂 ∈ 𝑃)
19		mideulem.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
20	19	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ 𝑃)
21		mideulem.5	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
22	21	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
23		mideulem.6	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
24	23	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
25		mideulem.7	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
26	25	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝐴𝐿𝐵))
27		mideulem.8	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
28	27	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝑄𝐼𝑂))
29		simplr 766	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ 𝑃)
30		simprl 768	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ (𝐵𝐼𝑄))
31		simprr 770	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴 − 𝑂) = (𝐵 − 𝑟))
32	2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 31	opphllem 27092	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))
33	1, 32	reximddv 3206	. 2 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
34		mideulem.9	. . 3 ⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄))
35		eqid 2740	. . . 4 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
36	2, 3, 4, 35, 6, 9, 17, 11, 15	legov 26942	. . 3 ⊢ (𝜑 → ((𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄) ↔ ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))))
37	34, 36	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟)))
38	33, 37	r19.29a 3220	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∃wrex 3067 class class class wbr 5079 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 distcds 16967 TarskiGcstrkg 26784 Itvcitv 26790 LineGclng 26791 ≤Gcleg 26939 pInvGcmir 27009 ⟂Gcperpg 27052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-oadd 8290 df-er 8479 df-map 8598 df-pm 8599 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-dju 9658 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-xnn0 12304 df-z 12318 df-uz 12580 df-fz 13237 df-fzo 13380 df-hash 14041 df-word 14214 df-concat 14270 df-s1 14297 df-s2 14557 df-s3 14558 df-trkgc 26805 df-trkgb 26806 df-trkgcb 26807 df-trkg 26810 df-cgrg 26868 df-leg 26940 df-mir 27010 df-rag 27051 df-perpg 27053

This theorem is referenced by: midex 27094

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