Theorem omeulem2 8547

Description: Lemma for omeu 8549: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
omeulem2	⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))

Proof of Theorem omeulem2

Step	Hyp	Ref	Expression
1		simp3l 1202	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐷 ∈ On)
2		eloni 6342	. . . . . 6 ⊢ (𝐷 ∈ On → Ord 𝐷)
3		ordsucss 7793	. . . . . 6 ⊢ (Ord 𝐷 → (𝐵 ∈ 𝐷 → suc 𝐵 ⊆ 𝐷))
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → suc 𝐵 ⊆ 𝐷))
5		simp2l 1200	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐵 ∈ On)
6		onsuc 7787	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → suc 𝐵 ∈ On)
8		simp1l 1198	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ∈ On)
9		simp1r 1199	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ≠ ∅)
10		on0eln0 6389	. . . . . . . 8 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
11	8, 10	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
12	9, 11	mpbird 257	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ∅ ∈ 𝐴)
13		omword 8534	. . . . . 6 ⊢ (((suc 𝐵 ∈ On ∧ 𝐷 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝐵 ⊆ 𝐷 ↔ (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
14	7, 1, 8, 12, 13	syl31anc 1375	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (suc 𝐵 ⊆ 𝐷 ↔ (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
15	4, 14	sylibd 239	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
16		omcl 8500	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
17	8, 1, 16	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o 𝐷) ∈ On)
18		simp3r 1203	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ 𝐴)
19		onelon 6357	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → 𝐸 ∈ On)
20	8, 18, 19	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ On)
21		oaword1 8516	. . . . . 6 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
22		sstr 3955	. . . . . . 7 ⊢ (((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) ∧ (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
23	22	expcom 413	. . . . . 6 ⊢ ((𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸) → ((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
24	21, 23	syl 17	. . . . 5 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
25	17, 20, 24	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
26	15, 25	syld 47	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
27		simp2r 1201	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
28		onelon 6357	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
29	8, 27, 28	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ On)
30		omcl 8500	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
31	8, 5, 30	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o 𝐵) ∈ On)
32		oaord 8511	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐴 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴)))
33	32	biimpa 476	. . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) ∧ 𝐶 ∈ 𝐴) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
34	29, 8, 31, 27, 33	syl31anc 1375	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
35		omsuc 8490	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
36	8, 5, 35	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
37	34, 36	eleqtrrd 2831	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵))
38		ssel 3940	. . 3 ⊢ ((𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸) → (((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
39	26, 37, 38	syl6ci 71	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
40		simpr 484	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ 𝐸)
41		oaord 8511	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐸 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
42	40, 41	imbitrid 244	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
43		oveq2 7395	. . . . . . 7 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
44	43	oveq1d 7402	. . . . . 6 ⊢ (𝐵 = 𝐷 → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
45	44	adantr 480	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
46	45	eleq2d 2814	. . . 4 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → (((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸) ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
47	42, 46	mpbidi 241	. . 3 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
48	29, 20, 31, 47	syl3anc 1373	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
49	39, 48	jaod 859	1 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ⊆ wss 3914  ∅c0 4296  Ord word 6331  Oncon0 6332  suc csuc 6334  (class class class)co 7387   +_o coa 8431   ·_o comu 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-oadd 8438  df-omul 8439

This theorem is referenced by:  omopth2  8548