dsmmacl - Metamath Proof Explorer

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Theorem dsmmacl 21295

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 2732	. . 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 2732	. . . . . 6 ⊢ (𝑆 ⊕_m 𝑅) = (𝑆 ⊕_m 𝑅)
9		dsmmcl.h	. . . . . 6 ⊢ 𝐻 = (Base‘(𝑆 ⊕_m 𝑅))
10	6	ffnd 6718	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
11	1, 8, 2, 9, 5, 10	dsmmelbas 21293	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝐽 ∈ (Base‘𝑃))
14		dsmmacl.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝐻)
15	1, 8, 2, 9, 5, 10	dsmmelbas 21293	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝐾 ∈ (Base‘𝑃))
18	1, 2, 3, 4, 5, 6, 13, 17	prdsplusgcl 18655	. 2 ⊢ (𝜑 → (𝐽 + 𝐾) ∈ (Base‘𝑃))
19	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑆 ∈ 𝑉)
20	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ 𝑊)
21	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑅 Fn 𝐼)
22	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐽 ∈ (Base‘𝑃))
23	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐾 ∈ (Base‘𝑃))
24		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼)
25	1, 2, 19, 20, 21, 22, 23, 3, 24	prdsplusgfval 17419	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝐽 + 𝐾)‘𝑎) = ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)))
26	25	neeq1d 3000	. . . 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 3439	. . 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 496	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
29	16	simprd 496	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
30		unfi 9171	. . . . 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 584	. . . 4 ⊢ (𝜑 → ({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∪ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))}) ∈ Fin)
32		neorian 3037	. . . . . . . . . 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 356	. . . . . . . 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 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑅‘𝑎) ∈ Mnd)
36		eqid 2732	. . . . . . . . . . 11 ⊢ (Base‘(𝑅‘𝑎)) = (Base‘(𝑅‘𝑎))
37		eqid 2732	. . . . . . . . . . 11 ⊢ (0_g‘(𝑅‘𝑎)) = (0_g‘(𝑅‘𝑎))
38	36, 37	mndidcl 18639	. . . . . . . . . 10 ⊢ ((𝑅‘𝑎) ∈ Mnd → (0_g‘(𝑅‘𝑎)) ∈ (Base‘(𝑅‘𝑎)))
39		eqid 2732	. . . . . . . . . . 11 ⊢ (+_g‘(𝑅‘𝑎)) = (+_g‘(𝑅‘𝑎))
40	36, 39, 37	mndlid 18644	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0_g‘(𝑅‘𝑎))(+_g‘(𝑅‘𝑎))(0_g‘(𝑅‘𝑎))) = (0_g‘(𝑅‘𝑎)))
42		oveq12 7417	. . . . . . . . . 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 2734	. . . . . . . . 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 2957	. . . . . 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 4305	. . . . 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 9266	. . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
51	27, 50	eqeltrd 2833	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
52	1, 8, 2, 9, 5, 10	dsmmelbas 21293	. 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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {crab 3432 ∪ cun 3946 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 Fincfn 8938 Basecbs 17143 +_gcplusg 17196 0_gc0g 17384 X_scprds 17390 Mndcmnd 18624 ⊕_m cdsmm 21285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-prds 17392 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-dsmm 21286

This theorem is referenced by: dsmmsubg 21297

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