Theorem lsmpropd 19924

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 1222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝜑)
2		simp12 1223	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ∈ 𝒫 𝐵)
3	2	elpwid 4569	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ⊆ 𝐵)
4		simp2 1155	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝑡)
5	3, 4	sseldd 3914	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝐵)
6		simp13 1224	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ 𝒫 𝐵)
7	6	elpwid 4569	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
8		simp3 1156	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
9	7, 8	sseldd 3914	. . . . . . 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 850	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
12	11	mpoeq3dva 7523	. . . . 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 5928	. . . 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 7523	. . 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 4577	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		mpoeq12 7519	. . . 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 596	. . 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 4577	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21		mpoeq12 7519	. . . 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 596	. . 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 2806	. 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 2763	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2763	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
27		eqid 2763	. . . 4 ⊢ (LSSum‘𝐾) = (LSSum‘𝐾)
28	25, 26, 27	lsmfval 19885	. . 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 2763	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
32		eqid 2763	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
33		eqid 2763	. . . 4 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	31, 32, 33	lsmfval 19885	. . 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 2808	1 ⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  𝒫 cpw 4560  ran crn 5657  ‘cfv 6561  (class class class)co 7446   ∈ cmpo 7448  Basecbs 17419  +_gcplusg 17460  LSSumclsm 19881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-lsm 19883

This theorem is used by:  hlhillsm  43216