Theorem lsmpropd 19750

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 1220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝜑)
2		simp12 1221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ∈ 𝒫 𝐵)
3	2	elpwid 4576	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ⊆ 𝐵)
4		simp2 1153	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝑡)
5	3, 4	sseldd 3946	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝐵)
6		simp13 1222	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ 𝒫 𝐵)
7	6	elpwid 4576	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
8		simp3 1154	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
9	7, 8	sseldd 3946	. . . . . . 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 849	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
12	11	mpoeq3dva 7491	. . . . 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 5932	. . . 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 7491	. . 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 4584	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		mpoeq12 7487	. . . 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 595	. . 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 4584	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21		mpoeq12 7487	. . . 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 595	. . 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 2813	. 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 2770	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2770	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
27		eqid 2770	. . . 4 ⊢ (LSSum‘𝐾) = (LSSum‘𝐾)
28	25, 26, 27	lsmfval 19711	. . 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 18	. 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 2770	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
32		eqid 2770	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
33		eqid 2770	. . . 4 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	31, 32, 33	lsmfval 19711	. . 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 18	. 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 2815	1 ⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  𝒫 cpw 4567  ran crn 5666  ‘cfv 6540  (class class class)co 7414   ∈ cmpo 7416  Basecbs 17272  +_gcplusg 17313  LSSumclsm 19707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-lsm 19709

This theorem is referenced by:  hlhillsm  42680