hdmapglem7 - Mathbox for Norm Megill

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Theorem hdmapglem7 42101

Description: Lemma for hdmapg 42102. 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 42099	. 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 42099	. 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 41282	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	8	lmodring 20810	. . . . . . . . . . . . . . 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 42061	. . . . . . . . . . . . 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 20176	. . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
34	1, 31, 32, 4, 5, 33, 2, 12	dvheveccl 41284	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
35	34	eldifad 3910	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
36	35	snssd 4762	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
37	1, 4, 5, 3	dochssv 41527	. . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . 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 3931	. . . . . . . . . . . . 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 42077	. . . . . . . . . . . 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 42066	. . . . . . . . . . 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 42067	. . . . . . . . . . . . 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 42098	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝐺‘𝑙)) = 𝑙)
48	47	oveq1d 7370	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝐺‘(𝐺‘𝑙)) × (𝐺‘𝑘)) = (𝑙 × (𝐺‘𝑘)))
49	46, 48	eqtrd 2768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝑘 × (𝐺‘𝑙))) = (𝑙 × (𝐺‘𝑘)))
50		eqid 2733	. . . . . . . . . . . . 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 42094	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘𝑣)‘𝑢)) = ((𝑆‘𝑢)‘𝑣))
53	49, 52	oveq12d 7373	. . . . . . . . . . 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 2768	. . . . . . . . . 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 42100	. . . . . . . . . . 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 6835	. . . . . . . . . 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 42100	. . . . . . . . . 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 2778	. . . . . . . . 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 6835	. . . . . . . . 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 6838	. . . . . . . 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 6835	. . . . . . 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 6835	. . . . . . . 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 6838	. . . . . . 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 2778	. . . . . 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 3189	. . . 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 3189	. 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 2113 ∃wrex 3057 ⊆ wss 3898 {csn 4577 ⟨cop 4583 I cid 5515 ↾ cres 5623 ‘cfv 6489 (class class class)co 7355 Basecbs 17127 +_gcplusg 17168 ._rcmulr 17169 Scalarcsca 17171 ·_𝑠 cvsca 17172 0_gc0g 17350 -_gcsg 18856 LSSumclsm 19554 Ringcrg 20159 LModclmod 20802 LSpanclspn 20913 HLchlt 39522 LHypclh 40156 LTrncltrn 40273 DVecHcdvh 41250 ocHcoch 41519 HDMapchdma 41964 HGMapchg 42055

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-riotaBAD 39125

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-tpos 8165 df-undef 8212 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-n0 12393 df-z 12480 df-uz 12743 df-fz 13415 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-sca 17184 df-vsca 17185 df-0g 17352 df-mre 17496 df-mrc 17497 df-acs 17499 df-proset 18208 df-poset 18227 df-plt 18242 df-lub 18258 df-glb 18259 df-join 18260 df-meet 18261 df-p0 18337 df-p1 18338 df-lat 18346 df-clat 18413 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-grp 18857 df-minusg 18858 df-sbg 18859 df-subg 19044 df-cntz 19237 df-oppg 19266 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-oppr 20264 df-dvdsr 20284 df-unit 20285 df-invr 20315 df-dvr 20328 df-nzr 20437 df-rlreg 20618 df-domn 20619 df-drng 20655 df-lmod 20804 df-lss 20874 df-lsp 20914 df-lvec 21046 df-lsatoms 39148 df-lshyp 39149 df-lcv 39191 df-lfl 39230 df-lkr 39258 df-ldual 39296 df-oposet 39348 df-ol 39350 df-oml 39351 df-covers 39438 df-ats 39439 df-atl 39470 df-cvlat 39494 df-hlat 39523 df-llines 39670 df-lplanes 39671 df-lvols 39672 df-lines 39673 df-psubsp 39675 df-pmap 39676 df-padd 39968 df-lhyp 40160 df-laut 40161 df-ldil 40276 df-ltrn 40277 df-trl 40331 df-tgrp 40915 df-tendo 40927 df-edring 40929 df-dveca 41175 df-disoa 41201 df-dvech 41251 df-dib 41311 df-dic 41345 df-dih 41401 df-doch 41520 df-djh 41567 df-lcdual 41759 df-mapd 41797 df-hvmap 41929 df-hdmap1 41965 df-hdmap 41966 df-hgmap 42056

This theorem is referenced by: hdmapg 42102

W3C validator