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Theorem 3oalem5 5556

Description: Lemma for 3OA (weak) orthoarguesian law.

**Hypotheses**
Ref	Expression
3oa.1	⊢ A ∈ C_ℋ
3oa.2	⊢ B ∈ C_ℋ
3oa.3	⊢ C ∈ C_ℋ
3oa.4	⊢ R = ((⊥ ‘B) ∩ (B ∨_ℋ A))
3oa.5	⊢ S = ((⊥ ‘C) ∩ (C ∨_ℋ A))

**Assertion**
Ref	Expression
3oalem5	⊢ ((B +_ℋ R) ∩ (C +_ℋ S)) = ((B ∨_ℋ R) ∩ (C ∨_ℋ S))

**Proof of Theorem 3oalem5**
Step	Hyp	Ref	Expression
1		3oa.4	. . . . 5 ⊢ R = ((⊥ ‘B) ∩ (B ∨_ℋ A))
2	1	3oalem4 5555	. . . 4 ⊢ R ⊆ (⊥ ‘B)
3		3oa.2	. . . . . . . 8 ⊢ B ∈ C_ℋ
4	3	chocl 5192	. . . . . . 7 ⊢ (⊥ ‘B) ∈ C_ℋ
5		3oa.1	. . . . . . . 8 ⊢ A ∈ C_ℋ
6	3, 5	chjcl 5379	. . . . . . 7 ⊢ (B ∨_ℋ A) ∈ C_ℋ
7	4, 6	chincl 5382	. . . . . 6 ⊢ ((⊥ ‘B) ∩ (B ∨_ℋ A)) ∈ C_ℋ
8	1, 7	eqeltr 1159	. . . . 5 ⊢ R ∈ C_ℋ
9	8, 3	osum 5538	. . . 4 ⊢ (R ⊆ (⊥ ‘B) → (R +_ℋ B) = (R ∨_ℋ B))
10	2, 9	ax-mp 6	. . 3 ⊢ (R +_ℋ B) = (R ∨_ℋ B)
11	3	chshi 5132	. . . 4 ⊢ B ∈ S_ℋ
12	8	chshi 5132	. . . 4 ⊢ R ∈ S_ℋ
13	11, 12	shscom 5333	. . 3 ⊢ (B +_ℋ R) = (R +_ℋ B)
14	3, 8	chjcom 5389	. . 3 ⊢ (B ∨_ℋ R) = (R ∨_ℋ B)
15	10, 13, 14	3eqtr4 1126	. 2 ⊢ (B +_ℋ R) = (B ∨_ℋ R)
16		3oa.5	. . . . 5 ⊢ S = ((⊥ ‘C) ∩ (C ∨_ℋ A))
17	16	3oalem4 5555	. . . 4 ⊢ S ⊆ (⊥ ‘C)
18		3oa.3	. . . . . . . 8 ⊢ C ∈ C_ℋ
19	18	chocl 5192	. . . . . . 7 ⊢ (⊥ ‘C) ∈ C_ℋ
20	18, 5	chjcl 5379	. . . . . . 7 ⊢ (C ∨_ℋ A) ∈ C_ℋ
21	19, 20	chincl 5382	. . . . . 6 ⊢ ((⊥ ‘C) ∩ (C ∨_ℋ A)) ∈ C_ℋ
22	16, 21	eqeltr 1159	. . . . 5 ⊢ S ∈ C_ℋ
23	22, 18	osum 5538	. . . 4 ⊢ (S ⊆ (⊥ ‘C) → (S +_ℋ C) = (S ∨_ℋ C))
24	17, 23	ax-mp 6	. . 3 ⊢ (S +_ℋ C) = (S ∨_ℋ C)
25	18	chshi 5132	. . . 4 ⊢ C ∈ S_ℋ
26	22	chshi 5132	. . . 4 ⊢ S ∈ S_ℋ
27	25, 26	shscom 5333	. . 3 ⊢ (C +_ℋ S) = (S +_ℋ C)
28	18, 22	chjcom 5389	. . 3 ⊢ (C ∨_ℋ S) = (S ∨_ℋ C)
29	24, 27, 28	3eqtr4 1126	. 2 ⊢ (C +_ℋ S) = (C ∨_ℋ S)
30	15, 29	ineq12i 1643	1 ⊢ ((B +_ℋ R) ∩ (C +_ℋ S)) = ((B ∨_ℋ R) ∩ (C ∨_ℋ S))

Colors of variables: wff set class

Syntax hints: = wceq 1091 ∈ wcel 1092 ∩ cin 1486 ⊆ wss 1487 ‘cfv 2422 (class class class)co 3001 C_ℋ cch 4968 ⊥cort 4969 +_ℋ cph 4970 ∨_ℋ chj 4972

This theorem is referenced by: 3oa 5558

This theorem was proved from axioms: ax-1 3 ax-2 4 ax-3 5 ax-mp 6 ax-4 673 ax-5 674 ax-6 675 ax-7 676 ax-gen 677 ax-8 798 ax-9 799 ax-10 800 ax-11 801 ax-12 802 ax-13 804 ax-14 805 ax-16 922 ax-17 925 ax-ext 1074 ax-rep 1075 ax-un 1076 ax-pow 1077 ax-reg 1078 ax-inf 1079 ax-ac 1080 ax-hilex 4983 ax-hvaddcl 4984 ax-hvcom 4985 ax-hvass 4986 ax-hvzercl 4987 ax-hvaddid 4988 ax-hvmulcl 4989 ax-hvmulid 4991 ax-hvmulass 4992 ax-hvdistr1 4993 ax-hvdistr2 4994 ax-hvmulzer 4995 ax-hicl 5043 ax-his1 5045 ax-his2 5046 ax-his3 5047 ax-his4 5048 ax-hcompl 5113

This theorem depends on definitions: df-bi 128 df-or 197 df-an 198 df-3or 582 df-3an 583 df-ex 679 df-sb 853 df-eu 1009 df-mo 1010 df-clab 1093 df-cleq 1097 df-clel 1099 df-ne 1192 df-ral 1205 df-rex 1206 df-reu 1207 df-rab 1208 df-v 1349 df-sbc 1441 df-dif 1489 df-un 1490 df-in 1491 df-ss 1492 df-pss 1494 df-nul 1708 df-if 1777 df-pw 1799 df-sn 1811 df-pr 1812 df-tp 1814 df-op 1815 df-uni 1920 df-int 1966 df-iun 1996 df-tr 2042 df-br 2063 df-opab 2098 df-eprel 2122 df-id 2125 df-po 2128 df-so 2138 df-sup 2154 df-fr 2169 df-we 2186 df-ord 2202 df-on 2203 df-lim 2204 df-suc 2205 df-om 2373 df-xp 2424 df-rel 2425 df-cnv 2426 df-co 2427 df-dm 2428 df-rn 2429 df-res 2430 df-ima 2431 df-fun 2432 df-fn 2433 df-f 2434 df-f1 2435 df-fo 2436 df-f1o 2437 df-fv 2438 df-rdg 2970 df-opr 3003 df-oprab 3004 df-1st 3087 df-2nd 3088 df-1o 3104 df-oadd 3106 df-omul 3107 df-er 3200 df-ec 3202 df-qs 3205 df-ni 3794 df-pli 3795 df-mi 3796 df-lti 3797 df-plpq 3829 df-mpq 3830 df-enq 3831 df-nq 3832 df-plq 3833 df-mq 3834 df-rq 3835 df-ltq 3836 df-1q 3837 df-np 3880 df-1p 3881 df-plp 3882 df-mp 3883 df-ltp 3884 df-plpr 3958 df-mpr 3959 df-enr 3960 df-nr 3961 df-plr 3962 df-mr 3963 df-ltr 3964 df-0r 3965 df-1r 3966 df-m1r 3967 df-c 4034 df-0 4035 df-1 4036 df-i 4037 df-r 4038 df-plus 4039 df-mul 4040 df-lt 4041 df-sub 4133 df-neg 4135 df-div 4216 df-le 4277 df-n 4423 df-2 4462 df-3 4463 df-4 4464 df-n0 4535 df-z 4564 df-seq 4661 df-exp 4676 df-sqr 4728 df-re 4790 df-im 4791 df-cj 4792 df-abs 4793 df-clim 4876 df-hvsub 4996 df-hnorm 5074 df-cauchy 5102 df-hlim 5107 df-sh 5114 df-ch 5127 df-oc 5156 df-ch0 5157 df-pj 5244 df-shsum 5275 df-chj 5277

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