dsmmacl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dsmmacl		Structured version Visualization version GIF version

Theorem dsmmacl 21682

Description: The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)

**Hypotheses**
Ref	Expression
dsmmcl.p	⊢ 𝑃 = (𝑆X_s𝑅)
dsmmcl.h	⊢ 𝐻 = (Base‘(𝑆 ⊕_m 𝑅))
dsmmcl.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
dsmmcl.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
dsmmcl.r	⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
dsmmacl.j	⊢ (𝜑 → 𝐽 ∈ 𝐻)
dsmmacl.k	⊢ (𝜑 → 𝐾 ∈ 𝐻)
dsmmacl.a	⊢ + = (+_g‘𝑃)

**Assertion**
Ref	Expression
dsmmacl	⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)

Proof of Theorem dsmmacl Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dsmmcl.p	. . 3 ⊢ 𝑃 = (𝑆X_s𝑅)
2		eqid 2728	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		dsmmacl.a	. . 3 ⊢ + = (+_g‘𝑃)
4		dsmmcl.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
5		dsmmcl.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
6		dsmmcl.r	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
7		dsmmacl.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐻)
8		eqid 2728	. . . . . 6 ⊢ (𝑆 ⊕_m 𝑅) = (𝑆 ⊕_m 𝑅)
9		dsmmcl.h	. . . . . 6 ⊢ 𝐻 = (Base‘(𝑆 ⊕_m 𝑅))
10	6	ffnd 6728	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
11	1, 8, 2, 9, 5, 10	dsmmelbas 21680	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ 𝐻 ↔ (𝐽 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)))
12	7, 11	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐽 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin))
13	12	simpld 493	. . 3 ⊢ (𝜑 → 𝐽 ∈ (Base‘𝑃))
14		dsmmacl.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝐻)
15	1, 8, 2, 9, 5, 10	dsmmelbas 21680	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ 𝐻 ↔ (𝐾 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)))
16	14, 15	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin))
17	16	simpld 493	. . 3 ⊢ (𝜑 → 𝐾 ∈ (Base‘𝑃))
18	1, 2, 3, 4, 5, 6, 13, 17	prdsplusgcl 18732	. 2 ⊢ (𝜑 → (𝐽 + 𝐾) ∈ (Base‘𝑃))
19	4	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑆 ∈ 𝑉)
20	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ 𝑊)
21	10	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑅 Fn 𝐼)
22	13	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐽 ∈ (Base‘𝑃))
23	17	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐾 ∈ (Base‘𝑃))
24		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼)
25	1, 2, 19, 20, 21, 22, 23, 3, 24	prdsplusgfval 17463	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝐽 + 𝐾)‘𝑎) = ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)))
26	25	neeq1d 2997	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ↔ ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎))))
27	26	rabbidva 3437	. . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} = {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎))})
28	12	simprd 494	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
29	16	simprd 494	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
30		unfi 9203	. . . . 5 ⊢ (({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin ∧ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin) → ({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∪ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))}) ∈ Fin)
31	28, 29, 30	syl2anc 582	. . . 4 ⊢ (𝜑 → ({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∪ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))}) ∈ Fin)
32		neorian 3034	. . . . . . . . . 10 ⊢ (((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))) ↔ ¬ ((𝐽‘𝑎) = (0_g‘(𝑅‘𝑎)) ∧ (𝐾‘𝑎) = (0_g‘(𝑅‘𝑎))))
33	32	bicomi 223	. . . . . . . . 9 ⊢ (¬ ((𝐽‘𝑎) = (0_g‘(𝑅‘𝑎)) ∧ (𝐾‘𝑎) = (0_g‘(𝑅‘𝑎))) ↔ ((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))))
34	33	con1bii 355	. . . . . . . 8 ⊢ (¬ ((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))) ↔ ((𝐽‘𝑎) = (0_g‘(𝑅‘𝑎)) ∧ (𝐾‘𝑎) = (0_g‘(𝑅‘𝑎))))
35	6	ffvelcdmda 7099	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑅‘𝑎) ∈ Mnd)
36		eqid 2728	. . . . . . . . . . 11 ⊢ (Base‘(𝑅‘𝑎)) = (Base‘(𝑅‘𝑎))
37		eqid 2728	. . . . . . . . . . 11 ⊢ (0_g‘(𝑅‘𝑎)) = (0_g‘(𝑅‘𝑎))
38	36, 37	mndidcl 18716	. . . . . . . . . 10 ⊢ ((𝑅‘𝑎) ∈ Mnd → (0_g‘(𝑅‘𝑎)) ∈ (Base‘(𝑅‘𝑎)))
39		eqid 2728	. . . . . . . . . . 11 ⊢ (+_g‘(𝑅‘𝑎)) = (+_g‘(𝑅‘𝑎))
40	36, 39, 37	mndlid 18721	. . . . . . . . . 10 ⊢ (((𝑅‘𝑎) ∈ Mnd ∧ (0_g‘(𝑅‘𝑎)) ∈ (Base‘(𝑅‘𝑎))) → ((0_g‘(𝑅‘𝑎))(+_g‘(𝑅‘𝑎))(0_g‘(𝑅‘𝑎))) = (0_g‘(𝑅‘𝑎)))
41	35, 38, 40	syl2anc2 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0_g‘(𝑅‘𝑎))(+_g‘(𝑅‘𝑎))(0_g‘(𝑅‘𝑎))) = (0_g‘(𝑅‘𝑎)))
42		oveq12 7435	. . . . . . . . . 10 ⊢ (((𝐽‘𝑎) = (0_g‘(𝑅‘𝑎)) ∧ (𝐾‘𝑎) = (0_g‘(𝑅‘𝑎))) → ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) = ((0_g‘(𝑅‘𝑎))(+_g‘(𝑅‘𝑎))(0_g‘(𝑅‘𝑎))))
43	42	eqeq1d 2730	. . . . . . . . 9 ⊢ (((𝐽‘𝑎) = (0_g‘(𝑅‘𝑎)) ∧ (𝐾‘𝑎) = (0_g‘(𝑅‘𝑎))) → (((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) = (0_g‘(𝑅‘𝑎)) ↔ ((0_g‘(𝑅‘𝑎))(+_g‘(𝑅‘𝑎))(0_g‘(𝑅‘𝑎))) = (0_g‘(𝑅‘𝑎))))
44	41, 43	syl5ibrcom 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (((𝐽‘𝑎) = (0_g‘(𝑅‘𝑎)) ∧ (𝐾‘𝑎) = (0_g‘(𝑅‘𝑎))) → ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) = (0_g‘(𝑅‘𝑎))))
45	34, 44	biimtrid 241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (¬ ((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))) → ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) = (0_g‘(𝑅‘𝑎))))
46	45	necon1ad 2954	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎)) → ((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎)))))
47	46	ss2rabdv 4073	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎))} ⊆ {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎)))})
48		unrab 4308	. . . . 5 ⊢ ({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∪ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))}) = {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎)) ∨ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎)))}
49	47, 48	sseqtrrdi 4033	. . . 4 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎))} ⊆ ({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∪ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))}))
50	31, 49	ssfid 9298	. . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
51	27, 50	eqeltrd 2829	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
52	1, 8, 2, 9, 5, 10	dsmmelbas 21680	. 2 ⊢ (𝜑 → ((𝐽 + 𝐾) ∈ 𝐻 ↔ ((𝐽 + 𝐾) ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)))
53	18, 51, 52	mpbir2and 711	1 ⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 {crab 3430 ∪ cun 3947 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 Basecbs 17187 +_gcplusg 17240 0_gc0g 17428 X_scprds 17434 Mndcmnd 18701 ⊕_m cdsmm 21672

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-dsmm 21673

This theorem is referenced by: dsmmsubg 21684

W3C validator