Theorem lsmpropd 19597

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 1200	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝜑)
2		simp12 1201	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ∈ 𝒫 𝐵)
3	2	elpwid 4606	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑡 ⊆ 𝐵)
4		simp2 1134	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝑡)
5	3, 4	sseldd 3978	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑥 ∈ 𝐵)
6		simp13 1202	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ 𝒫 𝐵)
7	6	elpwid 4606	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑢 ⊆ 𝐵)
8		simp3 1135	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
9	7, 8	sseldd 3978	. . . . . . 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 834	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵 ∧ 𝑢 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑢) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
12	11	mpoeq3dva 7482	. . . . 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 5931	. . . 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 7482	. . 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 4614	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17		mpoeq12 7478	. . . 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 583	. . 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 4614	. . . 4 ⊢ (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21		mpoeq12 7478	. . . 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 583	. . 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 2774	. 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 2726	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2726	. . . 4 ⊢ (+g‘𝐾) = (+g‘𝐾)
27		eqid 2726	. . . 4 ⊢ (LSSum‘𝐾) = (LSSum‘𝐾)
28	25, 26, 27	lsmfval 19558	. . 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 2726	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
32		eqid 2726	. . . 4 ⊢ (+g‘𝐿) = (+g‘𝐿)
33		eqid 2726	. . . 4 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	31, 32, 33	lsmfval 19558	. . 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 2776	1 ⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  𝒫 cpw 4597  ran crn 5670  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17153  +_gcplusg 17206  LSSumclsm 19554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-lsm 19556

This theorem is referenced by:  hlhillsm  41344