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Theorem prlngsymquadlem 29423

Description: Lemma for prlngsymquad 29424. (Contributed by Thierry Arnoux, 20-Jul-2026.)

**Hypotheses**
Ref	Expression
symquadprlng.p	⊢ 𝑃 = (Base‘𝐺)
symquadprlng.d	⊢ − = (dist‘𝐺)
symquadprlng.l	⊢ 𝐿 = (LineG‘𝐺)
symquadprlng.r	⊢ ∥ = (parlnG‘𝐺)
symquadprlng.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
symquadprlng.1	⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
symquadprlng.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
symquadprlng.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
symquadprlng.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
symquadprlng.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
prlngsymquad.2	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
prlngsymquad.3	⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
prlngsymquad.4	⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
prlngsymquadlem.t	⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)

**Assertion**
Ref	Expression
prlngsymquadlem	⊢ (𝜑 → 𝑇 = 𝑊)

**Proof of Theorem prlngsymquadlem**
Step	Hyp	Ref	Expression
1		symquadprlng.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2761	. 2 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		symquadprlng.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		symquadprlng.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		symquadprlng.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
6		symquadprlng.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7		symquadprlng.r	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
8		prlngsymquad.4	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
9	3, 7, 4, 8	prlngrcl2 29403	. . . . 5 ⊢ (𝜑 → (𝑊𝐿𝑋) ∈ ran 𝐿)
10	1, 2, 3, 4, 5, 6, 9	tglnne 29078	. . . 4 ⊢ (𝜑 → 𝑊 ≠ 𝑋)
11	1, 2, 3, 4, 5, 6, 10	tglinecom 29085	. . 3 ⊢ (𝜑 → (𝑊𝐿𝑋) = (𝑋𝐿𝑊))
12	11, 9	eqeltrrd 2862	. 2 ⊢ (𝜑 → (𝑋𝐿𝑊) ∈ ran 𝐿)
13		prlngsymquad.3	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
14	3, 7, 4, 13	prlngrcl2 29403	. 2 ⊢ (𝜑 → (𝑍𝐿𝑊) ∈ ran 𝐿)
15		symquadprlng.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16		symquadprlng.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
17		prlngsymquad.2	. . . 4 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
18	1, 2, 3, 4, 6, 15, 16, 5, 17	tglineneq 29095	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
19	4	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiG)
20	13	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
21		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
22	3, 7, 19, 20, 21	prlngin0 29404	. . . . 5 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)
23	1, 2, 3, 4, 6, 15, 16, 17	ncolne1 29075	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
24	1, 2, 3, 4, 6, 15, 23	tglinerflx1 29083	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑌))
25	24	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑌))
26	10	necomd 3011	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 𝑊)
27	1, 2, 3, 4, 6, 5, 26	tglinerflx1 29083	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑊))
28	27	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑊))
29		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑊) = (𝑍𝐿𝑊))
30	28, 29	eleqtrd 2863	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑍𝐿𝑊))
31	25, 30	elind 4146	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)))
32	31	ne0d 4288	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) ≠ ∅)
33	32	neneqd 2961	. . . . 5 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ¬ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)
34	22, 33	pm2.65da 829	. . . 4 ⊢ ((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) → ¬ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
35		nne 2960	. . . 4 ⊢ (¬ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊) ↔ (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
36	34, 35	sylib 221	. . 3 ⊢ ((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
37	18, 36	mteqand 3047	. 2 ⊢ (𝜑 → (𝑋𝐿𝑊) ≠ (𝑍𝐿𝑊))
38		prlngsymquadlem.t	. . . . . 6 ⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)
39		symquadprlng.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
40		eqid 2761	. . . . . . 7 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
41	1, 3, 2, 4, 15, 16, 6, 17	ncoltgdim2 29010	. . . . . . . 8 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
42	1, 39, 2, 4, 41, 6, 16	midcl 29264	. . . . . . 7 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) ∈ 𝑃)
43		eqid 2761	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍)) = ((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))
44	1, 39, 2, 3, 40, 4, 42, 43, 15	mircl 29115	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ∈ 𝑃)
45	38, 44	eqeltrid 2865	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
46	3, 7, 4, 8	prlngrcl1 29402	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
47	1, 2, 3, 4, 15, 16, 46	tglnne 29078	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
48	47	necomd 3011	. . . . . . 7 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
49	1, 40, 43, 4, 42, 16, 15	mirleqb 29148	. . . . . . . 8 ⊢ (𝜑 → (𝑍 = 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
50	49	necon3bid 3000	. . . . . . 7 ⊢ (𝜑 → (𝑍 ≠ 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
51	48, 50	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌))
52		eqidd 2762	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑋(midG‘𝐺)𝑍))
53	1, 39, 2, 4, 41, 6, 16, 40, 42	ismidb 29265	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ↔ (𝑋(midG‘𝐺)𝑍) = (𝑋(midG‘𝐺)𝑍)))
54	52, 53	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝑍 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋))
55	54	eqcomd 2767	. . . . . . 7 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) = 𝑍)
56	1, 39, 2, 3, 40, 4, 42, 43, 6, 55	mircom 29117	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) = 𝑋)
57	38	eqcomi 2770	. . . . . . 7 ⊢ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) = 𝑇
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) = 𝑇)
59	51, 56, 58	3netr3d 3032	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑇)
60	1, 2, 3, 4, 6, 45, 59	tglinerflx2 29084	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑇))
61	1, 40, 43, 4, 42, 6, 15	mirleqb 29148	. . . . . . . 8 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
62	61	necon3bid 3000	. . . . . . 7 ⊢ (𝜑 → (𝑋 ≠ 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
63	23, 62	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌))
64	63, 55, 58	3netr3d 3032	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ 𝑇)
65	1, 2, 3, 4, 16, 45, 64	tglinerflx2 29084	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑍𝐿𝑇))
66	60, 65	elind 4146	. . 3 ⊢ (𝜑 → 𝑇 ∈ ((𝑋𝐿𝑇) ∩ (𝑍𝐿𝑇)))
67		symquadprlng.1	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
68	1, 2, 3, 4, 15, 16, 47	tglinecom 29085	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
69		eqid 2761	. . . . . . 7 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
70		eqid 2761	. . . . . . 7 ⊢ (midG‘𝐺) = (midG‘𝐺)
71	1, 3, 2, 4, 15, 16, 6, 17	ncolcom 29006	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
72	71	orsild 1019	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑍𝐿𝑌))
73	6, 72	eldifd 3910	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
74	1, 39, 2, 4, 41, 16, 6	midcom 29269	. . . . . . . 8 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = (𝑋(midG‘𝐺)𝑍))
75	1, 39, 2, 4, 41, 15, 45, 40, 42	ismidb 29265	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ↔ (𝑌(midG‘𝐺)𝑇) = (𝑋(midG‘𝐺)𝑍)))
76	38, 75	mpbii 236	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑇) = (𝑋(midG‘𝐺)𝑍))
77	76	eqcomd 2767	. . . . . . . 8 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑇))
78	74, 77	eqtrd 2796	. . . . . . 7 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = (𝑌(midG‘𝐺)𝑇))
79	1, 3, 69, 7, 70, 4, 67, 16, 15, 73, 45, 78, 48	prlngmid2 29421	. . . . . 6 ⊢ (𝜑 → (𝑍𝐿𝑌) ∥ (𝑋𝐿𝑇))
80	68, 79	eqbrtrd 5127	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑋𝐿𝑇))
81	8, 11	breqtrd 5131	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑋𝐿𝑊))
82	1, 2, 3, 4, 6, 45, 59	tglinerflx1 29083	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑇))
83	1, 7, 4, 67, 80, 81, 82, 27	prlngeq 29417	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑇) = (𝑋𝐿𝑊))
84	1, 3, 2, 4, 15, 16, 6, 17	ncolrot2 29008	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
85	84	orsild 1019	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
86	16, 85	eldifd 3910	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
87	1, 3, 69, 7, 70, 4, 67, 6, 15, 86, 45, 77, 23	prlngmid2 29421	. . . . 5 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑇))
88	1, 2, 3, 4, 16, 45, 64	tglinerflx1 29083	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑇))
89	1, 2, 3, 4, 16, 5, 14	tglnne 29078	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
90	1, 2, 3, 4, 16, 5, 89	tglinerflx1 29083	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑊))
91	1, 7, 4, 67, 87, 13, 88, 90	prlngeq 29417	. . . 4 ⊢ (𝜑 → (𝑍𝐿𝑇) = (𝑍𝐿𝑊))
92	83, 91	ineq12d 4167	. . 3 ⊢ (𝜑 → ((𝑋𝐿𝑇) ∩ (𝑍𝐿𝑇)) = ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
93	66, 92	eleqtrd 2863	. 2 ⊢ (𝜑 → 𝑇 ∈ ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
94	1, 2, 3, 4, 6, 5, 26	tglinerflx2 29084	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑋𝐿𝑊))
95	1, 2, 3, 4, 16, 5, 89	tglinerflx2 29084	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑍𝐿𝑊))
96	94, 95	elind 4146	. 2 ⊢ (𝜑 → 𝑊 ∈ ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
97	1, 2, 3, 4, 12, 14, 37, 93, 96	tglineineq 29093	1 ⊢ (𝜑 → 𝑇 = 𝑊)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∩ cin 3898 ∅c0 4279 class class class wbr 5103 ran crn 5652 ‘cfv 6531 (class class class)co 7412 Basecbs 17367 distcds 17417 TarskiGcstrkg 28871 TarskiG_Ecstrkge 28876 Itvcitv 28877 LineGclng 28878 pInvGcmir 29106 hlGcplng 29233 midGcmid 29259 parlnGcprlng 29396

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-oadd 8464 df-er 8701 df-map 8833 df-pm 8834 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-n0 12588 df-xnn0 12661 df-z 12675 df-uz 12947 df-fz 13621 df-fzo 13769 df-hash 14455 df-word 14639 df-concat 14696 df-s1 14723 df-s2 14979 df-s3 14980 df-trkgc 28892 df-trkgb 28893 df-trkgcb 28894 df-trkge 28895 df-trkgld 28896 df-trkg 28897 df-cgrg 28956 df-ismt 28978 df-leg 29028 df-hlg 29046 df-mir 29107 df-rag 29151 df-perpg 29153 df-hpg 29218 df-plng 29234 df-mid 29261 df-lmi 29262 df-cgra 29297 df-prlng 29397

This theorem is used by: prlngsymquad 29424 prlngsymquadopp 29425

W3C validator