Theorem addsproplem6 27447

Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <s 𝑍)
addsproplem6.6	⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))

Assertion

Ref	Expression
addsproplem6	⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addspropord.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
2		addspropord.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ No )
3		addsproplem6.6	. . . 4 ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))
4		addspropord.5	. . . 4 ⊢ (𝜑 → 𝑌 <s 𝑍)
5		nodense 27184	. . . 4 ⊢ (((𝑌 ∈ No ∧ 𝑍 ∈ No ) ∧ (( bday ‘𝑌) = ( bday ‘𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
6	1, 2, 3, 4, 5	syl22anc 837	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
7		addsproplem.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
8		addspropord.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ No )
9	7, 8, 1	addsproplem3 27444	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})))
10	9	simp1d 1142	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No )
11	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No )
12	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
13	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑋 ∈ No )
14		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ No )
15		unidm 4151	. . . . . . 7 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
16		simprr1 1221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
17		bdayelon 27267	. . . . . . . . . 10 ⊢ ( bday ‘𝑚) ∈ On
18		bdayelon 27267	. . . . . . . . . 10 ⊢ ( bday ‘𝑌) ∈ On
19		bdayelon 27267	. . . . . . . . . 10 ⊢ ( bday ‘𝑋) ∈ On
20		naddel2 8683	. . . . . . . . . 10 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
21	17, 18, 19, 20	mp3an 1461	. . . . . . . . 9 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
22	16, 21	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
23		elun1 4175	. . . . . . . 8 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
25	15, 24	eqeltrid 2837	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
26	12, 13, 14, 14, 25	addsproplem1 27442	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
27	26	simpld 495	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No )
28		uncom 4152	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
29	28	eleq2i 2825	. . . . . . . . . . 11 ⊢ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) ↔ ((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))))
30	29	imbi1i 349	. . . . . . . . . 10 ⊢ ((((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
31	30	ralbii 3093	. . . . . . . . 9 ⊢ (∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32	31	2ralbii 3128	. . . . . . . 8 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
33	7, 32	sylib 217	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
34	33, 8, 2	addsproplem3 27444	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
35	34	simp1d 1142	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ No )
36	35	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No )
37	9	simp3d 1144	. . . . . 6 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
39		ovex 7438	. . . . . . 7 ⊢ (𝑋 +s 𝑌) ∈ V
40	39	snid 4663	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
41	40	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42		oldbday 27384	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
43	18, 14, 42	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
44	16, 43	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
45		simprr2 1222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46		breq2 5151	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (𝑌 <s 𝑎 ↔ 𝑌 <s 𝑚))
47		rightval 27348	. . . . . . . . . 10 ⊢ ( R ‘𝑌) = {𝑎 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑎}
48	46, 47	elrab2 3685	. . . . . . . . 9 ⊢ (𝑚 ∈ ( R ‘𝑌) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑚))
49	44, 45, 48	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( R ‘𝑌))
50		eqid 2732	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
51		oveq2 7413	. . . . . . . . 9 ⊢ (ℎ = 𝑚 → (𝑋 +s ℎ) = (𝑋 +s 𝑚))
52	51	rspceeqv 3632	. . . . . . . 8 ⊢ ((𝑚 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
53	49, 50, 52	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
54		ovex 7438	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) ∈ V
55		eqeq1 2736	. . . . . . . . 9 ⊢ (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ℎ) ↔ (𝑋 +s 𝑚) = (𝑋 +s ℎ)))
56	55	rexbidv 3178	. . . . . . . 8 ⊢ (𝑔 = (𝑋 +s 𝑚) → (∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ)))
57	54, 56	elab 3667	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
58	53, 57	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})
59		elun2 4176	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
60	58, 59	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
61	38, 41, 60	ssltsepcd 27284	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
62	33	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
63	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑍 ∈ No )
64	62, 13, 63	addsproplem3 27444	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
65	64	simp2d 1143	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
66	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑌) = ( bday ‘𝑍))
67	16, 66	eleqtrd 2835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑍))
68		bdayelon 27267	. . . . . . . . . . 11 ⊢ ( bday ‘𝑍) ∈ On
69		oldbday 27384	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
70	68, 14, 69	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
71	67, 70	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑍)))
72		simprr3 1223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
73		breq1 5150	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (𝑎 <s 𝑍 ↔ 𝑚 <s 𝑍))
74		leftval 27347	. . . . . . . . . 10 ⊢ ( L ‘𝑍) = {𝑎 ∈ ( O ‘( bday ‘𝑍)) ∣ 𝑎 <s 𝑍}
75	73, 74	elrab2 3685	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑍) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑚 <s 𝑍))
76	71, 72, 75	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( L ‘𝑍))
77		oveq2 7413	. . . . . . . . 9 ⊢ (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
78	77	rspceeqv 3632	. . . . . . . 8 ⊢ ((𝑚 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
79	76, 50, 78	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
80		eqeq1 2736	. . . . . . . . 9 ⊢ (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
81	80	rexbidv 3178	. . . . . . . 8 ⊢ (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
82	54, 81	elab 3667	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
83	79, 82	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)})
84		elun2 4176	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
85	83, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
86		ovex 7438	. . . . . . 7 ⊢ (𝑋 +s 𝑍) ∈ V
87	86	snid 4663	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
88	87	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
89	65, 85, 88	ssltsepcd 27284	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
90	11, 27, 36, 61, 89	slttrd 27251	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
91	6, 90	rexlimddv 3161	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
92	1, 8	addscomd 27440	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
93	2, 8	addscomd 27440	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
94	91, 92, 93	3brtr4d 5179	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ∪ cun 3945  {csn 4627   class class class wbr 5147  Oncon0 6361  ‘cfv 6540  (class class class)co 7405   +no cnadd 8660   No csur 27132   <s cslt 27133   bday cbday 27134   <<s csslt 27271   O cold 27327   L cleft 27329   R cright 27330   +_s cadds 27432

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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

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-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-0s 27314  df-made 27331  df-old 27332  df-left 27334  df-right 27335  df-norec2 27422  df-adds 27433

This theorem is referenced by:  addsproplem7  27448