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Theorem mideulem 28680

Description: Lemma for mideu 28682. 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 780	. . 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 726	. . . 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 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ∈ 𝑃)
11		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐵 ∈ 𝑃)
13		mideulem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
14	13	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ≠ 𝐵)
15		mideulem.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
16	15	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑄 ∈ 𝑃)
17		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
18	17	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑂 ∈ 𝑃)
19		mideulem.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
20	19	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ 𝑃)
21		mideulem.5	. . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
22	21	ad2antrr 726	. . . 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 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
25		mideulem.7	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
26	25	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝐴𝐿𝐵))
27		mideulem.8	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
28	27	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝑄𝐼𝑂))
29		simplr 768	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ 𝑃)
30		simprl 770	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ (𝐵𝐼𝑄))
31		simprr 772	. . . 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 28679	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))
33	1, 32	reximddv 3158	. 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 2734	. . . 4 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
36	2, 3, 4, 35, 6, 9, 17, 11, 15	legov 28529	. . 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 3149	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∃wrex 3059 class class class wbr 5123 ‘cfv 6541 (class class class)co 7413 Basecbs 17229 distcds 17282 TarskiGcstrkg 28371 Itvcitv 28377 LineGclng 28378 ≤Gcleg 28526 pInvGcmir 28596 ⟂Gcperpg 28639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-er 8727 df-map 8850 df-pm 8851 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-dju 9923 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-n0 12510 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13530 df-fzo 13677 df-hash 14352 df-word 14535 df-concat 14591 df-s1 14616 df-s2 14869 df-s3 14870 df-trkgc 28392 df-trkgb 28393 df-trkgcb 28394 df-trkg 28397 df-cgrg 28455 df-leg 28527 df-mir 28597 df-rag 28638 df-perpg 28640

This theorem is referenced by: midex 28681

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