hdmapglem7 - Mathbox for Norm Megill

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Theorem hdmapglem7 41974

Description: Lemma for hdmapg 41975. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our 𝐸, (𝑂‘{𝐸}), 𝑋, 𝑌, 𝑘, 𝑢, 𝑙, and 𝑣 correspond respectively to Baer's w, H, x, y, x', x'', y', and y'', and our ((𝑆‘𝑌)‘𝑋) corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapglem7.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapglem7.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapglem7.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapglem7.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapglem7.v	⊢ 𝑉 = (Base‘𝑈)
hdmapglem7.p	⊢ + = (+_g‘𝑈)
hdmapglem7.q	⊢ · = ( ·_𝑠 ‘𝑈)
hdmapglem7.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapglem7.b	⊢ 𝐵 = (Base‘𝑅)
hdmapglem7.a	⊢ ⊕ = (LSSum‘𝑈)
hdmapglem7.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmapglem7.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapglem7.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmapglem7.t	⊢ × = (._r‘𝑅)
hdmapglem7.z	⊢ 0 = (0_g‘𝑅)
hdmapglem7.c	⊢ ✚ = (+_g‘𝑅)
hdmapglem7.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapglem7.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hdmapglem7.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
hdmapglem7	⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))

Proof of Theorem hdmapglem7 Dummy variables 𝑘 𝑙 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapglem7.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapglem7.e	. . 3 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3		hdmapglem7.o	. . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
4		hdmapglem7.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		hdmapglem7.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
6		hdmapglem7.p	. . 3 ⊢ + = (+_g‘𝑈)
7		hdmapglem7.q	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
8		hdmapglem7.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑈)
9		hdmapglem7.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
10		hdmapglem7.a	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
11		hdmapglem7.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
12		hdmapglem7.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		hdmapglem7.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	hdmapglem7a 41972	. 2 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
15		hdmapglem7.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15	hdmapglem7a 41972	. 2 ⊢ (𝜑 → ∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣))
17		hdmapglem7.c	. . . . . . . . . . . 12 ⊢ ✚ = (+_g‘𝑅)
18		hdmapglem7.g	. . . . . . . . . . . 12 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
19	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20	1, 4, 12	dvhlmod 41155	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	8	lmodring 20802	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
22	20, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑅 ∈ Ring)
24		simplrr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑘 ∈ 𝐵)
25		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑙 ∈ 𝐵)
26	1, 4, 8, 9, 18, 19, 25	hgmapcl 41934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘𝑙) ∈ 𝐵)
27		hdmapglem7.t	. . . . . . . . . . . . . 14 ⊢ × = (._r‘𝑅)
28	9, 27	ringcl 20169	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ 𝐵 ∧ (𝐺‘𝑙) ∈ 𝐵) → (𝑘 × (𝐺‘𝑙)) ∈ 𝐵)
29	23, 24, 26, 28	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝑘 × (𝐺‘𝑙)) ∈ 𝐵)
30		hdmapglem7.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
31		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
34	1, 31, 32, 4, 5, 33, 2, 12	dvheveccl 41157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
35	34	eldifad 3914	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
36	35	snssd 4761	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
37	1, 4, 5, 3	dochssv 41400	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
38	12, 36, 37	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
39	38	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝑂‘{𝐸}) ⊆ 𝑉)
40		simplrl 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑢 ∈ (𝑂‘{𝐸}))
41	39, 40	sseldd 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑢 ∈ 𝑉)
42		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑣 ∈ (𝑂‘{𝐸}))
43	39, 42	sseldd 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑣 ∈ 𝑉)
44	1, 4, 5, 8, 9, 30, 19, 41, 43	hdmapipcl 41950	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝑆‘𝑣)‘𝑢) ∈ 𝐵)
45	1, 4, 8, 9, 17, 18, 19, 29, 44	hgmapadd 41939	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢))) = ((𝐺‘(𝑘 × (𝐺‘𝑙))) ✚ (𝐺‘((𝑆‘𝑣)‘𝑢))))
46	1, 4, 8, 9, 27, 18, 19, 24, 26	hgmapmul 41940	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝑘 × (𝐺‘𝑙))) = ((𝐺‘(𝐺‘𝑙)) × (𝐺‘𝑘)))
47	1, 4, 8, 9, 18, 19, 25	hgmapvv 41971	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝐺‘𝑙)) = 𝑙)
48	47	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝐺‘(𝐺‘𝑙)) × (𝐺‘𝑘)) = (𝑙 × (𝐺‘𝑘)))
49	46, 48	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝑘 × (𝐺‘𝑙))) = (𝑙 × (𝐺‘𝑘)))
50		eqid 2731	. . . . . . . . . . . . 13 ⊢ (-_g‘𝑈) = (-_g‘𝑈)
51		hdmapglem7.z	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑅)
52	1, 2, 3, 4, 5, 6, 50, 7, 8, 9, 27, 51, 30, 18, 19, 40, 42, 24, 24	hdmapglem5 41967	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘𝑣)‘𝑢)) = ((𝑆‘𝑢)‘𝑣))
53	49, 52	oveq12d 7364	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝐺‘(𝑘 × (𝐺‘𝑙))) ✚ (𝐺‘((𝑆‘𝑣)‘𝑢))) = ((𝑙 × (𝐺‘𝑘)) ✚ ((𝑆‘𝑢)‘𝑣)))
54	45, 53	eqtrd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢))) = ((𝑙 × (𝐺‘𝑘)) ✚ ((𝑆‘𝑢)‘𝑣)))
55	13	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → 𝑋 ∈ 𝑉)
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 55, 27, 51, 17, 30, 18, 42, 40, 25, 24	hdmapglem7b 41973	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)) = ((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢)))
57	56	fveq2d 6826	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = (𝐺‘((𝑘 × (𝐺‘𝑙)) ✚ ((𝑆‘𝑣)‘𝑢))))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 55, 27, 51, 17, 30, 18, 40, 42, 24, 25	hdmapglem7b 41973	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)) = ((𝑙 × (𝐺‘𝑘)) ✚ ((𝑆‘𝑢)‘𝑣)))
59	54, 57, 58	3eqtr4d 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
60	59	3adantl3 1169	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
61	60	3adant3 1132	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
62		simp3 1138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑌 = ((𝑙 · 𝐸) + 𝑣))
63	62	fveq2d 6826	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝑆‘𝑌) = (𝑆‘((𝑙 · 𝐸) + 𝑣)))
64		simp13 1206	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → 𝑋 = ((𝑘 · 𝐸) + 𝑢))
65	63, 64	fveq12d 6829	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → ((𝑆‘𝑌)‘𝑋) = ((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢)))
66	65	fveq2d 6826	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = (𝐺‘((𝑆‘((𝑙 · 𝐸) + 𝑣))‘((𝑘 · 𝐸) + 𝑢))))
67	64	fveq2d 6826	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝑆‘𝑋) = (𝑆‘((𝑘 · 𝐸) + 𝑢)))
68	67, 62	fveq12d 6829	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → ((𝑆‘𝑋)‘𝑌) = ((𝑆‘((𝑘 · 𝐸) + 𝑢))‘((𝑙 · 𝐸) + 𝑣)))
69	61, 66, 68	3eqtr4d 2776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) ∧ 𝑌 = ((𝑙 · 𝐸) + 𝑣)) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))
70	69	3exp 1119	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → ((𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵) → (𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))))
71	70	rexlimdvv 3188	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) ∧ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌)))
72	71	3exp 1119	. . 3 ⊢ (𝜑 → ((𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵) → (𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌)))))
73	72	rexlimdvv 3188	. 2 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢) → (∃𝑣 ∈ (𝑂‘{𝐸})∃𝑙 ∈ 𝐵 𝑌 = ((𝑙 · 𝐸) + 𝑣) → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))))
74	14, 16, 73	mp2d 49	1 ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 ⊆ wss 3902 {csn 4576 ⟨cop 4582 I cid 5510 ↾ cres 5618 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 -_gcsg 18848 LSSumclsm 19547 Ringcrg 20152 LModclmod 20794 LSpanclspn 20905 HLchlt 39395 LHypclh 40029 LTrncltrn 40146 DVecHcdvh 41123 ocHcoch 41392 HDMapchdma 41837 HGMapchg 41928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 38998

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19230 df-oppg 19259 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-oppr 20256 df-dvdsr 20276 df-unit 20277 df-invr 20307 df-dvr 20320 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20796 df-lss 20866 df-lsp 20906 df-lvec 21038 df-lsatoms 39021 df-lshyp 39022 df-lcv 39064 df-lfl 39103 df-lkr 39131 df-ldual 39169 df-oposet 39221 df-ol 39223 df-oml 39224 df-covers 39311 df-ats 39312 df-atl 39343 df-cvlat 39367 df-hlat 39396 df-llines 39543 df-lplanes 39544 df-lvols 39545 df-lines 39546 df-psubsp 39548 df-pmap 39549 df-padd 39841 df-lhyp 40033 df-laut 40034 df-ldil 40149 df-ltrn 40150 df-trl 40204 df-tgrp 40788 df-tendo 40800 df-edring 40802 df-dveca 41048 df-disoa 41074 df-dvech 41124 df-dib 41184 df-dic 41218 df-dih 41274 df-doch 41393 df-djh 41440 df-lcdual 41632 df-mapd 41670 df-hvmap 41802 df-hdmap1 41838 df-hdmap 41839 df-hgmap 41929

This theorem is referenced by: hdmapg 41975

W3C validator