hdmapglem7 - Mathbox for Norm Megill

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Theorem hdmapglem7 41896

Description: Lemma for hdmapg 41897. 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 41894	. 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 41894	. 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 41077	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	8	lmodring 20750	. . . . . . . . . . . . . . 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 41856	. . . . . . . . . . . . 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 20135	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝐾) = (Base‘𝐾)
32		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
33		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
34	1, 31, 32, 4, 5, 33, 2, 12	dvheveccl 41079	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
35	34	eldifad 3923	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
36	35	snssd 4769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
37	1, 4, 5, 3	dochssv 41322	. . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 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 41872	. . . . . . . . . . . 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 41861	. . . . . . . . . . 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 41862	. . . . . . . . . . . . 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 41893	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝐺‘𝑙)) = 𝑙)
48	47	oveq1d 7384	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → ((𝐺‘(𝐺‘𝑙)) × (𝐺‘𝑘)) = (𝑙 × (𝐺‘𝑘)))
49	46, 48	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘(𝑘 × (𝐺‘𝑙))) = (𝑙 × (𝐺‘𝑘)))
50		eqid 2729	. . . . . . . . . . . . 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 41889	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝑂‘{𝐸}) ∧ 𝑘 ∈ 𝐵)) ∧ (𝑣 ∈ (𝑂‘{𝐸}) ∧ 𝑙 ∈ 𝐵)) → (𝐺‘((𝑆‘𝑣)‘𝑢)) = ((𝑆‘𝑢)‘𝑣))
53	49, 52	oveq12d 7387	. . . . . . . . . . 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 2764	. . . . . . . . . 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 41895	. . . . . . . . . . 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 6844	. . . . . . . . . 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 41895	. . . . . . . . . 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 2774	. . . . . . . . 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 6844	. . . . . . . . 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 6847	. . . . . . . 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 6844	. . . . . . 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 6844	. . . . . . . 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 6847	. . . . . . 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 2774	. . . . . 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 3191	. . . 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 3191	. 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 1540 ∈ wcel 2109 ∃wrex 3053 ⊆ wss 3911 {csn 4585 ⟨cop 4591 I cid 5525 ↾ cres 5633 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 +_gcplusg 17196 ._rcmulr 17197 Scalarcsca 17199 ·_𝑠 cvsca 17200 0_gc0g 17378 -_gcsg 18843 LSSumclsm 19540 Ringcrg 20118 LModclmod 20742 LSpanclspn 20853 HLchlt 39316 LHypclh 39951 LTrncltrn 40068 DVecHcdvh 41045 ocHcoch 41314 HDMapchdma 41759 HGMapchg 41850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-riotaBAD 38919

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-undef 8229 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17380 df-mre 17523 df-mrc 17524 df-acs 17526 df-proset 18231 df-poset 18250 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-cntz 19225 df-oppg 19254 df-lsm 19542 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-nzr 20398 df-rlreg 20579 df-domn 20580 df-drng 20616 df-lmod 20744 df-lss 20814 df-lsp 20854 df-lvec 20986 df-lsatoms 38942 df-lshyp 38943 df-lcv 38985 df-lfl 39024 df-lkr 39052 df-ldual 39090 df-oposet 39142 df-ol 39144 df-oml 39145 df-covers 39232 df-ats 39233 df-atl 39264 df-cvlat 39288 df-hlat 39317 df-llines 39465 df-lplanes 39466 df-lvols 39467 df-lines 39468 df-psubsp 39470 df-pmap 39471 df-padd 39763 df-lhyp 39955 df-laut 39956 df-ldil 40071 df-ltrn 40072 df-trl 40126 df-tgrp 40710 df-tendo 40722 df-edring 40724 df-dveca 40970 df-disoa 40996 df-dvech 41046 df-dib 41106 df-dic 41140 df-dih 41196 df-doch 41315 df-djh 41362 df-lcdual 41554 df-mapd 41592 df-hvmap 41724 df-hdmap1 41760 df-hdmap 41761 df-hgmap 41851

This theorem is referenced by: hdmapg 41897

W3C validator