dsmmacl - Metamath Proof Explorer

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Theorem dsmmacl 21631

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 2726	. . 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 2726	. . . . . 6 ⊢ (𝑆 ⊕_m 𝑅) = (𝑆 ⊕_m 𝑅)
9		dsmmcl.h	. . . . . 6 ⊢ 𝐻 = (Base‘(𝑆 ⊕_m 𝑅))
10	6	ffnd 6711	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
11	1, 8, 2, 9, 5, 10	dsmmelbas 21629	. . . . 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 494	. . 3 ⊢ (𝜑 → 𝐽 ∈ (Base‘𝑃))
14		dsmmacl.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝐻)
15	1, 8, 2, 9, 5, 10	dsmmelbas 21629	. . . . 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 494	. . 3 ⊢ (𝜑 → 𝐾 ∈ (Base‘𝑃))
18	1, 2, 3, 4, 5, 6, 13, 17	prdsplusgcl 18695	. 2 ⊢ (𝜑 → (𝐽 + 𝐾) ∈ (Base‘𝑃))
19	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑆 ∈ 𝑉)
20	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐼 ∈ 𝑊)
21	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑅 Fn 𝐼)
22	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐽 ∈ (Base‘𝑃))
23	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝐾 ∈ (Base‘𝑃))
24		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼)
25	1, 2, 19, 20, 21, 22, 23, 3, 24	prdsplusgfval 17426	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝐽 + 𝐾)‘𝑎) = ((𝐽‘𝑎)(+_g‘(𝑅‘𝑎))(𝐾‘𝑎)))
26	25	neeq1d 2994	. . . 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 3433	. . 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 495	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
29	16	simprd 495	. . . . 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 583	. . . 4 ⊢ (𝜑 → ({𝑎 ∈ 𝐼 ∣ (𝐽‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∪ {𝑎 ∈ 𝐼 ∣ (𝐾‘𝑎) ≠ (0_g‘(𝑅‘𝑎))}) ∈ Fin)
32		neorian 3031	. . . . . . . . . 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 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑅‘𝑎) ∈ Mnd)
36		eqid 2726	. . . . . . . . . . 11 ⊢ (Base‘(𝑅‘𝑎)) = (Base‘(𝑅‘𝑎))
37		eqid 2726	. . . . . . . . . . 11 ⊢ (0_g‘(𝑅‘𝑎)) = (0_g‘(𝑅‘𝑎))
38	36, 37	mndidcl 18679	. . . . . . . . . 10 ⊢ ((𝑅‘𝑎) ∈ Mnd → (0_g‘(𝑅‘𝑎)) ∈ (Base‘(𝑅‘𝑎)))
39		eqid 2726	. . . . . . . . . . 11 ⊢ (+_g‘(𝑅‘𝑎)) = (+_g‘(𝑅‘𝑎))
40	36, 39, 37	mndlid 18684	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0_g‘(𝑅‘𝑎))(+_g‘(𝑅‘𝑎))(0_g‘(𝑅‘𝑎))) = (0_g‘(𝑅‘𝑎)))
42		oveq12 7413	. . . . . . . . . 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 2728	. . . . . . . . 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 2951	. . . . . 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 4068	. . . . 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 4300	. . . . 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 4028	. . . 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 2827	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)
52	1, 8, 2, 9, 5, 10	dsmmelbas 21629	. 2 ⊢ (𝜑 → ((𝐽 + 𝐾) ∈ 𝐻 ↔ ((𝐽 + 𝐾) ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ ((𝐽 + 𝐾)‘𝑎) ≠ (0_g‘(𝑅‘𝑎))} ∈ Fin)))
53	18, 51, 52	mpbir2and 710	1 ⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 ∪ cun 3941 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 Fincfn 8938 Basecbs 17150 +_gcplusg 17203 0_gc0g 17391 X_scprds 17397 Mndcmnd 18664 ⊕_m cdsmm 21621

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 7721 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 396 df-or 845 df-3or 1085 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-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 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-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-0g 17393 df-prds 17399 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-dsmm 21622

This theorem is referenced by: dsmmsubg 21633

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