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Theorem mideulem 28822

Description: Lemma for mideu 28824. 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 781	. . 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 727	. . . 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 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ∈ 𝑃)
11		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐵 ∈ 𝑃)
13		mideulem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
14	13	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ≠ 𝐵)
15		mideulem.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
16	15	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑄 ∈ 𝑃)
17		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
18	17	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑂 ∈ 𝑃)
19		mideulem.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
20	19	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ 𝑃)
21		mideulem.5	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
22	21	ad2antrr 727	. . . 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 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
25		mideulem.7	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
26	25	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝐴𝐿𝐵))
27		mideulem.8	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
28	27	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝑄𝐼𝑂))
29		simplr 769	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ 𝑃)
30		simprl 771	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ (𝐵𝐼𝑄))
31		simprr 773	. . . 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 28821	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))
33	1, 32	reximddv 3154	. 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 2737	. . . 4 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
36	2, 3, 4, 35, 6, 9, 17, 11, 15	legov 28671	. . 3 ⊢ (𝜑 → ((𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄) ↔ ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))))
37	34, 36	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟)))
38	33, 37	r19.29a 3146	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 class class class wbr 5086 ‘cfv 6494 (class class class)co 7362 Basecbs 17174 distcds 17224 TarskiGcstrkg 28513 Itvcitv 28519 LineGclng 28520 ≤Gcleg 28668 pInvGcmir 28738 ⟂Gcperpg 28781

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

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 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-oadd 8404 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-concat 14528 df-s1 14554 df-s2 14805 df-s3 14806 df-trkgc 28534 df-trkgb 28535 df-trkgcb 28536 df-trkg 28539 df-cgrg 28597 df-leg 28669 df-mir 28739 df-rag 28780 df-perpg 28782

This theorem is referenced by: midex 28823

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