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Theorem symquadprlnglem 28948

Description: Lemma for symquadprlnglem 28948. (Contributed by Thierry Arnoux, 20-Jul-2026.)

**Hypotheses**
Ref	Expression
symquadprlnglem.p	⊢ 𝑃 = (Base‘𝐺)
symquadprlnglem.d	⊢ − = (dist‘𝐺)
symquadprlnglem.l	⊢ 𝐿 = (LineG‘𝐺)
symquadprlnglem.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
symquadprlnglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
symquadprlnglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
symquadprlnglem.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
symquadprlnglem.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
symquadprlnglem.1	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
symquadprlnglem.2	⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
symquadprlnglem.3	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
symquadprlnglem.4	⊢ (𝜑 → 𝑌 ≠ 𝑊)
symquadprlnglem.5	⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))
symquadprlnglem.6	⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))

**Assertion**
Ref	Expression
symquadprlnglem	⊢ (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))

**Proof of Theorem symquadprlnglem**
Step	Hyp	Ref	Expression
1		symquadprlnglem.3	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
2		symquadprlnglem.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
3		symquadprlnglem.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
4		eqid 2763	. . . . . . 7 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
5		symquadprlnglem.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		symquadprlnglem.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7		symquadprlnglem.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8		symquadprlnglem.5	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))
9	2, 3, 4, 5, 6, 7, 8	tglngne 28797	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
10	9	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑋 ≠ 𝑍)
11		symquadprlnglem.d	. . . . . . . 8 ⊢ − = (dist‘𝐺)
12	5	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝐺 ∈ TarskiG)
13		symquadprlnglem.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
14		symquadprlnglem.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
15		symquadprlnglem.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
16	2, 4, 3, 5, 13, 14, 15	tgelrnln 28881	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌𝐿𝑊) ∈ ran 𝐿)
17		symquadprlnglem.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))
18	2, 3, 4, 5, 16, 17	tglnpt 28796	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
19	18	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑇 ∈ 𝑃)
20	7	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑍 ∈ 𝑃)
21	6	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑋 ∈ 𝑃)
22		eqid 2763	. . . . . . . . . . . 12 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
23		eqid 2763	. . . . . . . . . . . 12 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
24		symquadprlnglem.1	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
25		symquadprlnglem.2	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
26	8	orcd 886	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
27	17	orcd 886	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ (𝑌𝐿𝑊) ∨ 𝑌 = 𝑊))
28	2, 11, 4, 3, 22, 5, 23, 6, 13, 7, 14, 18, 1, 15, 24, 25, 26, 27	symquadlem 28944	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = (((pInvG‘𝐺)‘𝑇)‘𝑍))
29	28	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 − 𝑋) = (𝑇 − (((pInvG‘𝐺)‘𝑇)‘𝑍)))
30	2, 11, 4, 3, 22, 5, 18, 23, 7	mircgr 28912	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 − (((pInvG‘𝐺)‘𝑇)‘𝑍)) = (𝑇 − 𝑍))
31	29, 30	eqtr2d 2799	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 − 𝑍) = (𝑇 − 𝑋))
32	31	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → (𝑇 − 𝑍) = (𝑇 − 𝑋))
33		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑇 = 𝑍)
34	2, 11, 4, 12, 19, 20, 19, 21, 32, 33	tgcgreq 28729	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑇 = 𝑋)
35	34, 33	eqtr3d 2800	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → 𝑋 = 𝑍)
36		nne 2962	. . . . . 6 ⊢ (¬ 𝑋 ≠ 𝑍 ↔ 𝑋 = 𝑍)
37	35, 36	sylibr 237	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 = 𝑍) → ¬ 𝑋 ≠ 𝑍)
38	10, 37	pm2.65da 828	. . . 4 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → ¬ 𝑇 = 𝑍)
39	38	neqned 2965	. . 3 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝑇 ≠ 𝑍)
40	5	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝐺 ∈ TarskiG)
41	7	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝑍 ∈ 𝑃)
42	6	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝑋 ∈ 𝑃)
43	13	ad2antrr 738	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝑌 ∈ 𝑃)
44	18	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝑇 ∈ 𝑃)
45	5	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝐺 ∈ TarskiG)
46	7	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝑍 ∈ 𝑃)
47	13	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝑌 ∈ 𝑃)
48	18	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝑇 ∈ 𝑃)
49	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝑊 ∈ 𝑃)
50	2, 4, 3, 5, 13, 14, 15	tglinecom 28886	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌𝐿𝑊) = (𝑊𝐿𝑌))
51	17, 50	eleqtrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ (𝑊𝐿𝑌))
52	51	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → 𝑇 ∈ (𝑊𝐿𝑌))
53		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
54	2, 3, 4, 45, 46, 47, 49, 53	colcom 28805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → (𝑊 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
55	2, 4, 3, 45, 48, 49, 47, 46, 52, 54	coltr 28899	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → (𝑇 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
56	2, 3, 4, 45, 47, 46, 48, 55	colcom 28805	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → (𝑇 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
57	2, 3, 4, 45, 46, 47, 48, 56	colrot2 28807	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → (𝑌 ∈ (𝑇𝐿𝑍) ∨ 𝑇 = 𝑍))
58	57	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → (𝑌 ∈ (𝑇𝐿𝑍) ∨ 𝑇 = 𝑍))
59		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝑇 ≠ 𝑍)
60	59	neneqd 2963	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → ¬ 𝑇 = 𝑍)
61	58, 60	olcnd 890	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → 𝑌 ∈ (𝑇𝐿𝑍))
62	2, 3, 4, 5, 6, 7, 18, 26	colcom 28805	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
63	62	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → (𝑇 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
64	2, 4, 3, 40, 43, 44, 41, 42, 61, 63	coltr 28899	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
65	2, 3, 4, 40, 41, 42, 43, 64	colrot2 28807	. . 3 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) ∧ 𝑇 ≠ 𝑍) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
66	39, 65	mpdan 699	. 2 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
67	1, 66	mtand 827	1 ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ‘cfv 6538 (class class class)co 7412 Basecbs 17270 distcds 17320 TarskiGcstrkg 28674 Itvcitv 28680 LineGclng 28681 pInvGcmir 28907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-oadd 8458 df-er 8695 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-fz 13537 df-fzo 13685 df-hash 14369 df-word 14553 df-concat 14610 df-s1 14636 df-s2 14887 df-s3 14888 df-trkgc 28695 df-trkgb 28696 df-trkgcb 28697 df-trkg 28700 df-cgrg 28758 df-mir 28908

This theorem is referenced by: symquadmid 29086 symquadprlng 29190

W3C validator