Theorem lsmpropd 19695

Description: If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.) (Revised by AV, 25-Apr-2024.)

Hypotheses

Ref	Expression
lsmpropd.b1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lsmpropd.b2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
lsmpropd.p	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
lsmpropd.v1	⊢ (𝜑 → 𝐾 ∈ 𝑉)
lsmpropd.v2	⊢ (𝜑 → 𝐿 ∈ 𝑊)

Assertion

Ref	Expression
lsmpropd	⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem lsmpropd

Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1204	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝜑)
2		simp12 1205	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ∈ 𝒫 𝐵)
3	2	elpwid 4609	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ⊆ 𝐵)
4		simp2 1138	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝑡)
5	3, 4	sseldd 3984	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝐵)
6		simp13 1206	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ 𝒫 𝐵)
7	6	elpwid 4609	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
8		simp3 1139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
9	7, 8	sseldd 3984	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝐵)
10		lsmpropd.p	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
11	1, 5, 9, 10	syl12anc 837	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
12	11	mpoeq3dva 7510	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦)))
13	12	rneqd 5949	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦)))
14	13	mpoeq3dva 7510	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))))
15		lsmpropd.b1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
16	15	pweqd 4617	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		mpoeq12 7506	. . . 4 ⊢ ((𝒫 𝐵 = 𝒫 (Base‘𝐾) ∧ 𝒫 𝐵 = 𝒫 (Base‘𝐾)) → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))))
18	16, 16, 17	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))))
19		lsmpropd.b2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
20	19	pweqd 4617	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21		mpoeq12 7506	. . . 4 ⊢ ((𝒫 𝐵 = 𝒫 (Base‘𝐿) ∧ 𝒫 𝐵 = 𝒫 (Base‘𝐿)) → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))))
22	20, 20, 21	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))))
23	14, 18, 22	3eqtr3d 2785	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))))
24		lsmpropd.v1	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
25		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2737	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
27		eqid 2737	. . . 4 ⊢ (LSSum‘𝐾) = (LSSum‘𝐾)
28	25, 26, 27	lsmfval 19656	. . 3 ⊢ (𝐾 ∈ 𝑉 → (LSSum‘𝐾) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))))
29	24, 28	syl 17	. 2 ⊢ (𝜑 → (LSSum‘𝐾) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐾)𝑦))))
30		lsmpropd.v2	. . 3 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
31		eqid 2737	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
32		eqid 2737	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
33		eqid 2737	. . . 4 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	31, 32, 33	lsmfval 19656	. . 3 ⊢ (𝐿 ∈ 𝑊 → (LSSum‘𝐿) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))))
35	30, 34	syl 17	. 2 ⊢ (𝜑 → (LSSum‘𝐿) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝐿)𝑦))))
36	23, 29, 35	3eqtr4d 2787	1 ⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  𝒫 cpw 4600  ran crn 5686  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17247  +_gcplusg 17297  LSSumclsm 19652

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-lsm 19654

This theorem is referenced by:  hlhillsm  41962